text:
"excited about expanding its offshore wind presence"
"A lease auction of about 600 square miles acres off the coast of Massachusetts for the development of offshore wind energy has drawn a record $405 million in winning bids.
If fully developed, the three parcels could generate about 4.1 gigawatts of energy — enough to power nearly 1.5 million homes.
Interior Secretary Ryan Zinke said Friday the auction was the nation's eighth competitive lease sale for the development of offshore wind. He said it smashed the previous highest grossing offshore wind lease sale record of $42 million for an area off New York in 2016.
The provisional winners of the auction were Equinor Wind US, Mayflower Wind Energy and Vineyard Wind. They bid $135 million each for three similarly sized parcels of approximately 130,000 acres."
https://www.milforddailynews.com/news/20181214/auction-of-wind-parcels-off-mass-coast-draws-record-405m-in-bids
Image from Equinor's web page
More about Equinor Wind US https://www.equinor.com/en/news/us-wind-dec2018.html
More about Mayflower Wind Energy LLC
https://www.bloomberg.com/research/stocks/private/snapshot.asp?privcapId=279756673
More about Vineyard Wind https://www.vineyardwind.com/
Labels: Milford Daily News, Wind power, wind turbine


Network monitoring made affordable for any company size.
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ActiveXpets also claims that users of ActiveXperts Network Monitor 7.3 can run the software on a central computer without the need to install agents on the monitored computers.


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STK1040 48V 40W POWER AMPL ?
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STK4040 II - PMC/SAN - ?
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STK4048 II 86V 150W POWER AMPL ?
STK405-090A - originál Pioneer - ?


text:
Bei der Sächsisch-karolingischen Siedlung von Liebenau handelt es sich um eine im 8. und 9. Jahrhundert bestehende Siedlung im heutigen Liebenau in Niedersachsen. Die seit 2015 anhaltende Ausgrabung des Areals liefert Erkenntnisse zu sächsischen Siedlungen in karolingischer Zeit im  Mittelwesergebiet.

Lage
Die frühere Siedlungsstelle liegt am südlichen Ortsrand von Liebenau auf einer in West-Ost-Richtung verlaufenen Binnendüne zwischen der Weser und der Großen Aue. Der Siedlungsplatz wurde zu einem unbekannten Zeitpunkt aufgegeben. Sandüberwehungen konservierten seine Bodenreste und einstigen Laufhorizonte. Da das heute bewaldete Gelände nie intensiv landwirtschaftlich genutzt wurde, bestanden für die Siedlungsreste gute Erhaltungsbedingungen.

Nach dem Zweiten Weltkrieg wurde die Düne zur Sandgewinnung allmählich abgetragen. Da beim Sandabbau Keramik-, Knochen- und Schlackefunde zutage traten, wurde 1971 eine Sondagegrabung auf dem Areal durchgeführt. Sie erfolgte im Zuge der Ausgrabungen des etwa zwei Kilometer westlich liegenden Altsächsischen Gräberfeldes Liebenau. Anschließend wurde der Sandabbau beendet und das Gelände unter Denkmalschutz gestellt.

Ausgrabungen
Bei der Sondagegrabung von 1971 und einer weiteren Grabung von 1973 wurden drei nahe nebeneinander stehende Grubenhäuser entdeckt. Bei der Grabung geborgene Fundstücke waren eine Scheibenfibel, ein Kamm, ein Messer, ein Schlüssel, Spinnwirtel, Webgewichte und Schalen von Süßwassermuscheln. Eine Abfallgrube war mit Eisenschlacken verfüllt. Mit 350 Gefäßscherben war das keramische Fundaufkommen hoch.

2015 kam es zu einer dreiwöchigen Lehrgrabung des  Seminars für Ur- und Frühgeschichte der Universität Göttingen mit ehrenamtlichen Grabungshelfern im Rahmen des Projektes "ehrenWERT" der Klosterkammer Hannover. Die Ausgrabung erfolgte unter der Leitung des Archäologen Tobias Scholz in Kooperation mit der Kommunalarchäologie der Schaumburger Landschaft und Kommunalbehörden. Dabei wurden Pfostenstellungen von Gebäuden sowie Vorrats- und Abfallgruben gefunden. Dokumentierte Spuren von Wagenrädern deuteten auf einen einstigen Verkehrsweg hin. Zu den Funden gehörten Keramikfragmente, Metallschlacken, Brandlehm, Muschelschalen und Spinnwirtel. Durch eine Glasperle, eine Messerklinge, eine Pfeilspitze und eine kissenförmigen Rechteckfibel erfolgte die Datierung der Fundstelle in das späte 8. und frühe 9. Jahrhundert. Die Funde von Keramikscherben aus dem 8. und 9. Jahrhundert belaufen sich auf über 10.000 Stücke.

Weitere Ausgrabungen auf dem früheren Siedlungsareal fanden in den Jahren 2017, 2018 und 2019 durch die Universität Göttingen in Zusammenarbeit mit dem Verein RAUZWI – Lebendige Archäologie Mittelweser statt. Bei der Ausgrabung im Jahr 2018 wurde die Pfostenreihe eines Langhauses freigelegt. 2019 fanden sich 160 kg an Schlacken und 60 kg an Metallobjekten.

Grabungshelfer, die im Jahr 2015 an der Ausgrabung teilgenommen hatten, gründeten 2016 den Verein RAUZWI – Lebendige Archäologie Mittelweser. 2022 eröffnete der Verein in Liebenau eine Ausstellung  zur Siedlung unter dem Titel "Auf Sand gebaut - Von Sand begraben".

Bewertung
Die Bewohner des weilerartigen Siedlungsplatzes, deren Anzahl zwischen 30 und 100 geschätzt wird, waren Ackerbauern, Viehzüchter und Fischer. Sie nutzten die nahen Flüsse zum Fischfang sowie zum Sammeln essbarer Muscheln. Auch betrieben sie Eisenverarbeitung. Der für die Fundstelle zuständige Kommunalarchäologe der Schaumburger Landschaft Daniel Lau sieht in der Siedlung einen Handelsposten.

Literatur
 Tobias Scholz, Gundula Tessendorf: Auf Sand gebaut – von Sand begraben. Ausgrabung einer frühmittelalterlichen Siedlung im Umfeld des Gräberfeldes von Liebenau/Steyerberg. (= Schriften des RAUZWI-Vereins 1.), Nienburg/Delligsen, 2020.

Weblinks

 Siedlungsforschung und Landschaftsentwicklung Liebenau bei Universität Göttingen
 Ronald Reimann: Archäologische Grabung sächsisches Gräberfeld Liebenau:
 Teil 1: Die Befunde, Video bei Youtube (14:41 Minuten)
 Teil 2: Die Funde,  Video bei Youtube (14:26 Minuten)
 Teil 3: Der Grabungsablauf,  Video bei Youtube (08:07 Minuten)

Einzelnachweise

Liebenau (Niedersachsen)
Archäologischer Fundplatz in Niedersachsen
Archäologischer Fundplatz (Mittelalter)
Altsachsen
Wüstung im Landkreis Nienburg/Weser

redpajama_set_name: RedPajamaWikipedia


text:
Bizzare places to visit in the US
Lopinator
https://www.buzzfeed.com/eviecarrick/uni...o-visit-us
Ben & Jerry's Flavor Graveyard — Waterbury, Vermont
Carhenge — Alliance, Nebraska
Time Travel Mart — Los Angeles, California
Vent Haven Museum — Fort Mitchell, Kentucky
Bizzare and down right wtf
Always a clean chop.
Albrecht Bear
RE: Bizzare places to visit in the US
"At the Mütter Museum in Philly you can see the tumor from President Grover Cleveland's jaw, the skeleton of a very tall man, and the fused bones of Harry Eastlack, a man who died of fibrodysplasia ossificans progressiva, a rare disorder where tissues (like tendons and ligaments) gradually turn to bone and make the body immobile. The real showstopper, though, is the brain of Albert Einstein, which is laid out in slivers."
Ok, I'm hooked.
It is a fine thing to be honest, but it is also very important to be right. - Winston Churchill
The bare necessities of life
Agricultural Crash Monument, Norway, Illinois
https://www.roadsideamerica.com/story/2093
Pancho and Lefty
I just heard of this place and I want to go so bad.
They serve incredible food and have a few of the best Rollercoasters
Holiday World Theme Park
in Indiana. Where every day is Thanksgiving
https://www.yahoo.com/lifestyle/inside-h...03418.html
Bicnarok
Lopinator Wrote: (11-29-2021 06:45 PM)
You yanks are bizzare anyway, an andearing quality.
Bicnarok Wrote: (11-29-2021 08:04 PM)
I am British the Confeds will not be happy down south I tell ya
Yo check out obscura.com. Most of these on your list can be found there and there are even more
Haymarket Martyrs' Monument - Joe Hill 100 tour stop.
https://www.youtube.com/watch?v=2OQsv1I8VuM
The Haymarket Martyrs' Monument is a funeral monument sculpture located at Forest Home Cemetery in Forest Park, Illinois, a suburb of Chicago. Dedicated in 1893, it commemorates the defendants involved in the labor unrest and bombing related to the Haymarket Affair (1886). On February 18, 1997, it was designated a National Landmark.
tourist attractions attract wealthy foreigners more than americans.
The M65 Atomic Cannon - Rock Island Arsenal, Illinois
https://www.youtube.com/watch?v=fA9uSr8box8
The 280 MM Gun At The Nevada Proving Ground
https://www.youtube.com/watch?v=B9F-l_3eLcE&t=0s
Courtesy National Nuclear Security Administration / Nevada Site Office
The first artillery test was on May 25, 1953 at the Nevada Test Site. Fired as part of Operation Upshot-Knothole and codenamed Shot GRABLE, a 280 mm (11 inch) shell with a gun-type fission warhead was fired 10,000 m (6.2 miles) and detonated 160 m (525 ft) above the ground with an estimated yield of 15 kilotons. This was the only nuclear artillery shell ever actually fired in the US test program. The shell was 1384 mm (4.5 ft) long and weighed 365 kg (805 lb); it was fired from a special artillery piece, nicknamed "Atomic Annie", built by the Artillery Test Unit of Fort Sill, Oklahoma. Around 3,200 personnel were present. The warhead was designated the W9 and 80 were produced from 1952-53 for the T-124 shell. It was retired in 1957.
https://en.wikipedia.org/wiki/Nuclear_artillery
(This post was last modified: 11-29-2021 08:41 PM by silversides.)
Laserz
We went to Dollywood right before Thanksgiving to see the Christmas lights.
It was a nice family park with rides and entertainment, but it seems odd that she would name an amusement park after herself.
The Christmas lights were beautiful and the food was great!
You can't go back and change the beginning, but you can start where you are and change the ending. ~ C.S. Lewis
Merry Christmas from the bizarre place of Dollywood.
McDonald's Big Mac Museum, Pennsylvania, USA
Ever wondered about the origins of your Big Mac? No, neither have we – but in case you do want to know where it all began, there's a museum just for you.
The Big Mac Museum in Pennsylvania was founded by Jim Delligatti, the franchisee who opened Western Pennsylvania's first McDonald's restaurant. He also created the Big Mac, an item which first went on sale in his restaurant in 1967, with a price tag of just 45 cents.
Visitors can learn all about the brand's history and browse the collection of McDonald's memorabilia, which includes the world's largest (albeit plastic) Big Mac.
Uppity Me
Laserz Wrote: (11-29-2021 09:42 PM)
Cool, I had no idea that Dolly Parton had a theme park
Quote: Les pauvres n'ont-ils des Euros ?
Alors laissez les pauvres 'sh*t' dans les rues
Luvapottamus
The Fremont Troll(Seattle Fremont Neighborhood)
https://fremont.com/explore/sights/troll/
That's a real VW beetle(or super-beetle) under its left hand.(To give it scale)
(This post was last modified: 11-30-2021 12:08 AM by Luvapottamus.)

redpajama_set_name: RedPajamaCommonCrawl


text:
\section*{Introduction}

Currently, there is a growing interest in the study of population dynamics, mainly due to the need to have better control of epidemics, or to reduce the economic, biological and social losses caused by invasive species, among other needs.

The first mathematical studies aimed at describing interactions between populations took place in the 1920s \cite{intro}. During this period, the Lotka-Volterra mathematical model emerged \cite{lotka, volterra}. This model describes the predator-prey interaction between two populations through the system of ordinary differential equations,
\begin{eqnarray}
\frac{dS_1}{dt} &=& a_1S_1 - c_1S_1S_2 \nonumber \\ \\
\frac{dS_2}{dt} &=& -a_2S_2 + c_2S_1S_2, \nonumber
\label{1}
\end{eqnarray}
\noindent
where $S_1 > 0$ and $S_2 > 0$ denote the population densities of the interacting species, with $S_1$ being the density of prey and $S_2$ the density of predators. Furthermore, $a_1 > 0$ is the birth rate of the species $S_1$, $a_2 > 0$ is the death rate of the species $S_2$, and the parameters $c_1 > 0$ and $c_2 > 0$ are the interaction rates between the two species. The non-derivative terms on the right side of (1) are called reactive terms.

Later in the 1950s, C. S. Holling carried out experiments to investigate how the rate of prey capture by a predator is related to the density of the prey \cite{holling1,holling2}, a relationship called the functional response.

Holling identified three general categories of functional responses (Figure \ref{holl}). The type 1 functional response is linear, when the number of prey consumed increases in direct proportion to prey density. The type 2 functional response says that as the prey population increases, predators become satiated and consume a constant number of prey (saturation). The type 3 functional response is similar to type 2, except at low prey density, when prey switching effects occur \cite{Tansky}.

\begin{figure}[ht]
    \begin{center}
    \caption{The three types of functional responses identified by Holling.}
    \includegraphics[width=8cm]{hollin31.png}
    \label{holl}
    \end{center}
\end{figure}

Still in the 1950s, Carl B. Huffaker investigated the effects of spatial structure on the interaction of mite populations. Among several results, Huffaker showed that the predator-prey interaction could not survive in a homogeneous environment without dispersion \cite{huff1}.

Currently, through more complex mathematical modeling, there are works that emphasize the studies of the dynamics of invasive species, epidemics and other biological phenomena \cite{Deroos,pp3,pp1,petro1,pp4,cirilo1}.

As the description of the effects of predation on one species by another improved, the mathematical models became more complex. Thus, in parallel with the development of more realistic and complex biological models, studies on the convergence of numerical methods applied to these models have become essential.

The Lax Equivalence Theorem is fundamental for the analysis of the convergence of numerical solutions of PDEs. The theorem says that in a well-posed initial value problem that has been consistently discretized, stability of the numerical method is a necessary and sufficient condition for numerical convergence \cite{lalax}.

About stability of numerical methods, it is associated with the propagation of numerical errors in the iterative process \cite{hhh}. The numerical method is said to be stable if the errors decrease along the iterative process, otherwise, if the errors increase, the numerical solution will diverge and the numerical method is said to be unstable. Finally, a numerical method is said to be conditionally stable if it depends on certain parameters so that the errors remain limited. In the case of linear problems, the Von Neumann stability analysis is widely used  \cite{cuminato1999discretizaccao}. In the case of nonlinear initial condition problems, the stability of a numerical method can hardly be verified analytically \cite{est1}. In such situations, numerical experiments are used to analyze the stability of the numerical method used \cite{natti2}.

About consistency of numerical method, we say that a discretized problem is consistent when it tends to the original differential equation if the increments in the independent variables tend to zero \cite{consite1}.

In this context, the objective of this work is to verify the consistency and stability of a numerical method applied to a Telegraph Predator-Prey system. The conditions for the convergence of the numerical model are analyzed through numerical experiments.

\section*{Modeling the Telegraph Predator-Prey system}

In this section, the Maxwell-Cattaneo Diffusion equation system is first modeled and then the Telegraph Predator-Prey equation system is developed.

\subsection*{Maxwell-Cattaneo Diffusion system}

The classical law of diffusion, Fick's law \cite{paul2014fick}, is a law that describes the diffusion of a property in systems that are not in equilibrium. In systems where there are concentration gradients of a property, then there is a flow of this property which tends to homogenize the system.This homogenizing flow will go in the opposite direction of the gradient and, if this flow is weak, it can be approximated by the first term of a Taylor series, resulting in Fick's law. In the case of a Predator-Prey system in one dimension, this modeling is expressed mathematically by \cite{paul2014fick},
\begin{equation}\label{equ1leidefick}
J_{j} = -D_{j} \frac{\partial S_{j}}{\partial x},
\end{equation}
where flow $J_1$ is due to diffusion of prey $S_1$ and flow $J_2$ is due to diffusion of predators $S_2$, while $D_1$ and $D_2$ are the diffusivity coefficients for prey and predators, respectively. Note that the negative sign in \ref{equ1leidefick} indicates that the homogenizing flow occurs in the opposite direction of the concentration gradient, ie, from high concentrations to low concentrations.

On the other hand, Fick's law proposes that signals propagate with infinite speed, which in practice does not happen, configuring the so-called "Paradox of Fourier's law" \cite{mickens}. To correct this problem, Fick's law is modified by the Maxwell-Cattaneo Diffusion law \cite{cata} which introduces a lag time $\tau$ for each action. Thus, due to the finite speed of propagation of information, the system does not react instantly to an action.

For predator-prey systems, the Maxwell-Cattaneo Diffusion law is given by \cite{cirilo1}
 \begin{equation}
	\displaystyle \left(1+ \tau_{j} \frac{\partial }{\partial t}\right)J_{j} = -D_{j} \frac{\partial S_{j}}{\partial x},
	\label{EQ:M-C1}
      \end{equation}
where $\tau_1$ is the reaction time of the prey when exposed to predation and $\tau_2$ is the predator's reaction time to capture prey. Note that, contrary to the equation system (1), in the system (3) we have $S_j = S_j(x,t)$, for $j=1.2$, so that now the population densities of prey and predator, respectively, also depend on the spatial variable $x$.

\subsection*{Telegraph Predator-Prey system}

For predator-prey systems, in the context of Huffaker's hypotheses, the Maxwell-Cattaneo diffusion law must be incorporated into the population density conservation law. According to the Conservation Principle, the rate of change of a property in a volume $V$ must be equal to the net flux of that property through the surface of $V$, plus the amount of the property transformed into the interior of V due to reactive effects \cite{de2000tecnicas}. Thus, the conservation equation for predator-prey system, considering diffusive, reactive and delay effects, is written as
 \begin{equation}
	\displaystyle \frac{\partial S_j}{\partial t} = -\frac{\partial J_j}{\partial x} + F_j\left(S_j\right),
	\label{EQ:R}
      \end{equation}
where $J_j$ are the population flows described by (\ref{EQ:M-C1}) and $F_j(t,x,S)$ represents the prey reaction term, if $j=1$, and the predator reaction term, if $j=2$. Note that the equation (\ref{EQ:R}) is a transport equation.

Therefore, deriving \ref{EQ:R} with respect to time $t$ and \ref{EQ:M-C1} with respect to coordinate $x$, we have
\begin{equation}
	\displaystyle \frac{\partial^{2} S_j}{\partial t^{2}} = -\frac{\partial}{\partial t}\left(\frac{\partial J_j}{\partial x}\right) + \frac{\partial}{\partial t}F_j\left(S_j\right)
	\label{EQ:R-D2}
      \end{equation}

\begin{equation}
	\displaystyle \tau_j\frac{\partial}{\partial x}\left(\frac{\partial J_j}{\partial t}\right) +\frac{\partial J_j}{\partial x} = -D_j\frac{\partial^{2} S_j}{\partial x^{2}}.
	\label{EQ:M-C2}
      \end{equation}
\noindent
Noting that $F_j = F_j(S_j,x,t)$ and $S_j = S_j(x,t)$, it follows from the chain rule that
\begin{equation}
	\displaystyle \frac{\partial}{\partial t}F_j\left(S_j\right) = \frac{d}{d S} F_j(S_j)\hspace{1.5mm} \frac{\partial S_j}{\partial t},
	\label{EQ:R-DO2}
      \end{equation}
so that \ref{EQ:R-D2} is rewritten as
\begin{equation}
	\displaystyle \frac{\partial^{2} S_j}{\partial t^{2}} = -\frac{\partial}{\partial t}\left(\frac{\partial J_j}{\partial x}\right) + \frac{d}{d S_j} F_j(S_j)\hspace{1.5mm}\frac{\partial S_j}{\partial t}.
	\label{EQ:R-D3}
      \end{equation}
\noindent
Multiplying the equation \ref{EQ:R-D3} by $\tau_j$ and subtracting it from the equation \ref{EQ:M-C2}, we get that
 \begin{equation}
	\displaystyle \tau_j\frac{\partial^2 S_j}{\partial t^2} -
	              \tau_j \frac{d}{d S_j}F_j\left(S_j\right)\hspace{1.5mm} \frac{\partial S_j}{\partial t} =
	              D_j\frac{\partial^2 S_j}{\partial x^2} + \frac{\partial J_j}{\partial x}.
	\label{EQ:RT}
      \end{equation}
Finally, obtaining $\frac{\partial J_j}{\partial x}$ from \ref{EQ:R} and substituting in \ref{EQ:RT}, we obtain the delayed predator-prey equations, whose population densities are subject to reactive-diffusive processes \cite{equacao1,cirilo1},i.e.,
  \begin{equation}
	\displaystyle \tau_j\frac{\partial^2 S_j}{\partial t^2} +
	              \left[ 1 - \tau_j \frac{d}{d S_j}F_j\left(S_j\right)\right]\frac{\partial S_j}{\partial t} =
	              D_j\frac{\partial^2 S_j}{\partial x^2} + F_j\left(S_j\right).
	\label{EQ:R-T}
      \end{equation}
The equations \ref{EQ:R-T} will be designated in the remaining of the work as Telegraph Predator-Prey equations. Note that the equations \ref{EQ:R-T} have the same structure as the Telegraph equation, which was derived by William Thomson (Lord Kelvin) to describe the propagation of electrical signals in long conducting cables.

\section*{Discretization of the Telegraph Predator-Prey system}

Consider the equations \ref{EQ:R-T} with the following initial and boundary conditions
\begin{equation}
	\tau_{1}\frac{\partial^2 S_{1}}{\partial t^2}  +
	\left[ 1 - \tau_{1}\frac{d F_{1}}{d S_{1}}\right]\frac{\partial S_{1}}{\partial t} =  D_{1}\frac{\partial^2 S_{1}}{\partial x^2} + F_{1}\label{EQ:R_T_C}
\end{equation}
\begin{equation}
	\tau_{2}\frac{\partial^2 S_{2}}{\partial t^2}  +
	\left[ 1 - \tau_{2} \frac{d F_{2}}{d S_{2}}\right]\frac{\partial S_{2}}{\partial t} =  D_{2}\frac{\partial^2 S_{2}}{\partial x^2} + F_{2}\label{EQ:R_T_C_1}
\end{equation}
\begin{equation}
	S_{j}(x,0)=S_{j}^{0} \; \; , \; \; \frac{\partial S_{j}(x,t)}{\partial t}\bigg|^{t=0} = 0 \; \; , \; \; \forall x\in[0,L] \label{condicaoinicial}
\end{equation}
\begin{equation}
	S_{j}(0,t)=S_{j}(L,t) = 0 \; \; , \; \; \forall t\in[0,T] \; \; , \; \; j=1,2, \label{condicaodecontorno}
\end{equation}
where $t$ and $x$ are the temporal and spatial variables, $\tau_{1}$ and $\tau_{2}$ are the delay parameters of the populations, $D_{1}$ and $D_{2} $ are the diffusivity coefficients of the populations, $S_{1}(x,t)$ and $S_{2}(x,t)$ are the population densities, finally $F_{1}$ and $F_{2} $ are the prey and predator reactive terms of the populations, respectively.
In this work, it is considered that the reactive terms present functional responses of the Holling 1 type (see Figure 1), i.e.,
\begin{equation}\label{termoreativopresa}
	F_{1} = F_{1}(S_{1},S_{2}) = a_{1}S_{1} - b_{1}S_{1}^{2} - c_{1}S_{1}S_{2}
\end{equation}
\begin{equation}\label{termoreativopredador}
	F_{2} = F_{2}(S_{1},S_{2}) = - a_{2}S_{2} + c_{2}S_{1}S_{2},
\end{equation}
where $a_{1}$ is the prey birth rate, $b_{1}$ is the prey saturation term, $c_{1}$ is the prey death rate due to predation, $a_{2 }$ is the predator death rate in the absence of prey and $c_{2}$ is the predator birth rate due to predation \cite{natti1}.

The initial conditions $S_{1}^{0}$ and $S_{2}^{0}$ are the initial population densities of prey and predators, respectively. Finally, the  Dirichlet boundary conditions (14) impose that at the boundary of the problem domain, the population densities are null.

To discretize the Telegraph Predator-Prey equations (11-
12) the one-dimensional mesh shown in Figure \ref{deslocada} is used.
\begin{figure*}[ht]
	\begin{center}
	\caption{Discrete one-dimensional mesh used for the
Telegraph Predator-Prey system.}
		\includegraphics[width=13cm]{KARISTON1.png}
\label{deslocada} \vspace{4pt}
\end{center}
\end{figure*}

Cardinal point notation is used in the discretization of the problem domain. The labels $P$, $W$ and $E$ stand for center ($P$ point where the calculation is performed), east and west, respectively. Lowercase acronyms are cardinal variations from the center $P$. As shown in Figure \ref{deslocada}, the population densities $S_1$ and $S_2$ are calculated at the center of the cell. Note that storage shifted to population densities has a positive impact on numerical computation, reducing numerical instability \cite{paolo1,de2000tecnicas,paolo2,barba1,tati1}.

As for the numerical method for discretizing the PDEs, the Finite Difference Technique was used  \cite{cuminato1999discretizaccao,burden2008analise}. Thus, central finite differences are used for the discretization of the second derivatives, that is,
\begin{eqnarray}
\left.\left(\frac{\partial^{2}S_{j}}{\partial t^{2}}\right)\right|_{P}^{k}  \approx \frac{1}{(\Delta t)^{2}}\left(S_{j}|_{P}^{k + 1} - 2S_{j}|_{P}^{k} + S_{j}|_{P}^{k - 1}\right)
\label{eqsjtt}
\end{eqnarray}
\vspace{-0.4cm}
\begin{equation}
\label{eqsjxx}
\left.\left(\frac{\partial^{2}S_{j}}{\partial x^{2}}\right)\right|_{P}^{k} \approx \frac{1}{(\Delta x)^{2}}\left(\left. S_{j}\right|_{E}^{k} - 2\left.S_{j}\right|_{P}^{k} + \left.S_{j}\right|_{W}^{k} \right),
\end{equation}
while for the first derivative in time, regressive finite differences are used
\begin{equation}\label{eqsjt}
\left.\left(\frac{\partial}{\partial t}S_{j}\right)\right|_{P}^{k} \approx \frac{1}{\Delta t}\left(\left.S_{j}\right|_{P}^{k} - \left.S_{j}\right|_{P}^{k-1}\right).
\end{equation}

Finally, for the equation \ref{EQ:R_T_C}, it is still necessary to discretize the term
\begin{eqnarray}
\left.\left(\left[ 1 - \tau_{1} \frac{\partial}{\partial S_{1}}F_{1}\left(S_{1},S_{2}\right)\right]\frac{\partial S_{1}}{\partial t}\right)\right|_{P}^{k} =&& \nonumber \\
\left.\left(\frac{\partial S_{1}}{\partial t}\right)\right|_{P}^{k} - \tau_{1}\left.\left(\frac{\partial}{\partial S_{1}}F_{1}\left(S_{1},S_{2}\right)\right)\right|_{P}^{k}\left.\left(\frac{\partial S_{1}}{\partial t}\right)\right|_{P}^{k} . &&
 \label{eqs1f1t}
\end{eqnarray}
From \ref{termoreativopresa}, it follows that
\begin{equation}\label{eqs1f1t_2}
\left.\left(\frac{\partial}{\partial S_{1}}F_{1}\left(S_{1},S_{2}\right)\right)\right|_{P}^{k} = a_{1} - 2b_{1}\left.S_{1}\right|_{P}^{k} - c_{1}\left.S_{2}\right|_{P}^{k}, \end{equation}
then using the equations \ref{eqsjt} and \ref{eqs1f1t_2} in the equation \ref{eqs1f1t}, one obtains that
\begin{eqnarray}
\left.\left(\frac{\partial}{\partial S_{1}}F_{1}\left(S_{1},S_{2}\right)\frac{\partial S_{1}}{\partial t}\right)\right|_{P}^{k}  \approx
\nonumber \\
\left(a_{1} - 2b_{1}\left.S_{1}\right|_{P}^{k} - c_{1}\left.S_{2}\right|_{P}^{k}\right)\left(\frac{1}{\Delta t}\left(\left.S_{1}\right|_{P}^{k} - \left.S_{1}\right|_{P}^{k-1}\right)\right)
\approx \nonumber\\
\frac{1}{\Delta t}\left(a_{1}\left.S_{1}\right|_{P}^{k} - 2b_{1}\left.S_{1}^{2}\right|_{P}^{k} - c_{1}\left.S_{1}\right|_{P}^{k}\left.S_{2}\right|_{P}^{k}\right.\nonumber\\
- \left. a_{1}\left.S_{1}\right|_{P}^{k-1} + 2b_{1}\left.S_{1}\right|_{P}^{k-1}\left.S_{1}\right|_{P}^{k} + c_{1}\left.S_{1}\right|_{P}^{k-1}\left.S_{2}\right|_{P}^{k}\right).
\label{eqs1f1t_3}
\end{eqnarray}
Substituting (15), \ref{eqsjtt}, \ref{eqsjxx}, \ref{eqsjt}, and \ref{eqs1f1t_3} in \ref{EQ:R_T_C}, it follows that the discretization of the prey equation is given by
\begin{eqnarray}
 S_1|_{P}^{k + 1}  &=&  \Omega_{1}\left( \Upsilon_{1}S_{1}|_{W}^{k} + \Pi_{1}S_{1}^{2}|_{P}^{k} \right.\nonumber \\
 &+& \left.\Phi_{1}S_{1}|_{P}^{k} + \Lambda_{1}S_{1}|_{E}^{k} + \Gamma_{1}\right),
\label{finalpresa}
\end{eqnarray}
where
\begin{eqnarray}
\Upsilon_{1}  = - \frac{D_1}{{\Delta x}^2} \;\;\;\;,\;\;\;\;
\Lambda_{1} = - \frac{D_1}{{\Delta x}^2}
\end{eqnarray}
\begin{eqnarray}
\Omega_1 = - \frac{{\Delta t}^2}{\tau_1} \;\;\;\;,\;\;\;\;
\Pi_{1}  =  \frac{2\tau_1b_1}{\Delta t} + b_{1}
\end{eqnarray}
\begin{eqnarray}
\Phi_{1} & = &  \frac{1 - \tau_1\left(a_1 - c_1S_{2}|_{P}^{k} +2b_1S_{1}|_{P}^{k-1}\right)}{\Delta t}\nonumber\\
&-& \frac{2\tau_1}{{\Delta t}^2}
+ \frac{2D_1}{{\Delta x}^2} - a_{1} + c_{1}S_{2}|_{P}^{k}
\end{eqnarray}
\begin{eqnarray}
\Gamma_{1}=\frac{\tau_1S_1|_{P}^{k - 1}}{{\Delta t}^2} - \frac{\left(1 - \tau_1\left.(a_{1} - c_{1}S_{2}|_{P}^{k})\right.\right)S_{1}|_{P}^{k - 1}}{\Delta t}.
\end{eqnarray}
Note that the discretized prey equation \ref{finalpresa} is explicit in time and implicit in space.

In a similar way, the formula for the discretization of the predator equation is obtained. The only difference is in the derivation of the predator reactive term. From equation (16)
\begin{equation}\label{eqs1f2t_2}
\left.\left(\frac{\partial}{\partial S_{2}}F_{2}\left(S_{1},S_{2}\right)\right)\right|_{P}^{k} = -a_{2} + c_{2}\left.S_{1}\right|_{P}^{k}.
\end{equation}
So the equation for discretizing the predator equation is given by
\begin{eqnarray}
S_2|_{P}^{k + 1}  =  \Omega_{2}\left( \Upsilon_{2}S_{2}|_{W}^{k}  +\Phi_{2}S_{2}|_{P}^{k} + \Lambda_{2}S_{2}|_{E}^{k} + \Gamma_{2}\right),\label{finalpredador}
\end{eqnarray}
where
\begin{eqnarray}
\Upsilon_{2} = - \frac{D_2}{{\Delta x}^2} \;\;\;\;,\;\;\;\;
\Lambda_{2} = - \frac{D_2}{{\Delta x}^2} \;\;\;\;,\;\;\;\;
\Omega_2 = - \frac{{\Delta t}^2}{\tau_2}
\end{eqnarray}
\begin{eqnarray}
\Phi_{2} & = & \frac{1 - \tau_2(-a_2 + c_2S_1|_{P}^{k})}{\Delta t}
\nonumber\\
&-& \frac{2\tau_2}{{\Delta t}^2} + \frac{2D_2}{{\Delta x}^2}
+ a_2 - c_2S_1|_{P}^{k}
\end{eqnarray}
\begin{eqnarray}
\Gamma_{2}=\frac{\tau_2S_2|_{P}^{k - 1}}{{\Delta t}^2} - \frac{\left(1 - \tau_2(-a_2 +c_2S_1|_{P}^{k})\right)S_2|_{P}^{k -1}}{\Delta t}.
\end{eqnarray}
Again, note that the discretized prey equation \ref{finalpredador} is explicit in time and implicit in space.

\section*{Numerical model consistency}

The numerical solution of a problem does not always tend to the exact solution of the same problem. The resulting error between the exact solution of the continuous problem and the numerical solution obtained from the discretized equations is called Local Truncation Error (LTE) \cite{cuminato1999discretizaccao}. In most of the applied problems, the exact solution is not known and this is our case. In such situations one can estimate the Local Truncation Error through Taylor Series \cite{burden2008analise} and use this estimate to prove that the method is consistent at the limit of the continuum, when $\Delta x,\Delta t\to 0$.

Consider EDPs \ref{EQ:R_T_C} and \ref{EQ:R_T_C_1}. Using the Finite Difference method, we have the following discretized equations
\begin{eqnarray}
\frac{\tau_1}{\Delta t^{2}}\left(S_{1}\bigg|_{P}^{k + 1}  - 2S_{1}\bigg|_{P}^{k} + S_{1}\bigg|_{P}^{k - 1}\right)
\;\;\;\;\;\;\;\;\;\;\;\;\;
\nonumber\\
+ \frac{1}{\Delta t}\left[\left(1 - \tau_{1}\left(a_{1} - c_{1}\left.S_{2}\right|_{P}^{k} - 2b_{1}\left.S_{1}\right|_{P}^{k}\right)\right)S_{1}|_{P}^{k}\right.
\nonumber\\
- \left.\left(1 - \tau_{1}\left(a_{1} - c_{1}\left.S_{2}\right|_{P}^{k} - 2b_{1}\left.S_{1}\right|_{P}^{k}\right)\right)\left.S_{1}\right|_{P}^{k-1}\right]
\nonumber\\
- \frac{D_1}{\Delta x^{2}}\left( S_{1}\bigg|_{E}^{k} - 2S_{1}\bigg|_{P}^{k} + S_{1}\bigg|_{W}^{k} \right) - F_1\bigg|_P^{k} = 0
\label{final1}
\end{eqnarray}
and
\begin{eqnarray}
\frac{\tau_2}{\Delta t^{2}}\left(S_{2}\bigg|_{P}^{k + 1} - 2S_{2}\bigg|_{P}^{k} + S_{2}\bigg|_{P}^{k - 1}\right)
\;\;\;\;\;\;\;\;\;\;\;
\nonumber\\
+ \frac{1}{\Delta t}\left[\left(1 - \tau_{2}\left(- a_{2} + c_{2}\left.S_{1}\right|_{P}^{k}\right)\right)S_{2}|_{P}^{k}\right.
\;\;\;\;\;\;
\nonumber\\
- \left.\left(1 - \tau_{2}\left(- a_{2} + c_{2}\left.S_{1}\right|_{P}^{k}\right)\right)S_{2}|_{P}^{k-1}\right]
\;\;\;\;\;\;
\nonumber\\
- \frac{D_2}{\Delta x^{2}}\left( S_{2}\bigg|_{E}^{k} - 2S_{2}\bigg|_{P}^{k} + S_{2}\bigg|_{W}^{k} \right) - F_2\bigg|_P^{k} = 0,
\label{final2}
\end{eqnarray}
where $F_1$ and $F_2$ are the prey and predator reactive terms, equations (15) and (16), respectively.

Assuming that $W = P - \Delta x$ and $E = P + \Delta x$, let the Taylor Series expansions of the population densities in (33) and (34) be calculated around $k$ and $P $, that is,
\begin{equation}
\label{analise_consistencia_s1_termo_final1}
	S_{j}\bigg|_{P}^{k + 1} = S_{j}\bigg|_{P}^{k} + \Delta t\frac{\partial S_{j}}{\partial t}\bigg|_{P}^{k} + \frac{(\Delta t)^{2}}{2}\frac{\partial^{2} S_{j}}{\partial t^{2}}\bigg|_{P}^{k} + \frac{(\Delta t)^{3}}{3!}\frac{\partial^{3} S_{j}}{\partial t^{3}}\bigg|_{P}^{k} + \mathcal{O}(\Delta t^{4})
\end{equation}
\begin{equation}
\label{analise_consistencia_s1_termo_final2}
	S_{j}\bigg|_{P}^{k - 1} = S_{j}\bigg|_{P}^{k} - \Delta t\frac{\partial S_{j}}{\partial t}\bigg|_{P}^{k} + \frac{(\Delta t)^{2}}{2}\frac{\partial^{2} S_{j}}{\partial t^{2}}\bigg|_{P}^{k} - \frac{(\Delta t)^{3}}{3!}\frac{\partial^{3} S_{j}}{\partial t^{3}}\bigg|_{P}^{k} + \mathcal{O}(\Delta t^{4})
\end{equation}
\begin{equation}
\label{analise_consistencia_s1_termo_final3}
	S_{j}\bigg|_{W}^{k} = S_{j}\bigg|_{P}^{k} - \Delta x\frac{\partial S_{j}}{\partial x}\bigg|_{P}^{k} + \frac{(\Delta x)^{2}}{2}\frac{\partial^{2} S_{j}}{\partial x^{2}}\bigg|_{P}^{k} - \frac{(\Delta x)^{3}}{3!}\frac{\partial^{3} S_{j}}{\partial x^{3}}\bigg|_{P}^{k} + \mathcal{O}(\Delta x^{4})
\end{equation}
\begin{equation}
\label{analise_consistencia_s1_termo_final4}
	S_{j}\bigg|_{E}^{k} = S_{j}\bigg|_{P}^{k} + \Delta x\frac{\partial S_{j}}{\partial x}\bigg|_{P}^{k} + \frac{(\Delta x)^{2}}{2}\frac{\partial^{2} S_{j}}{\partial x^{2}}\bigg|_{P}^{k} + \frac{(\Delta x)^{3}}{3!}\frac{\partial^{3} S_{j}}{\partial x^{3}}\bigg|_{P}^{k} + \mathcal{O}(\Delta x^{4})
\end{equation}

\noindent Substituting the equations \ref{analise_consistencia_s1_termo_final1}- \ref{analise_consistencia_s1_termo_final4} in the equations \ref{final1} and \ref{final2}, with their respective $j = 1, 2$, we obtain that
\begin{eqnarray}
\underbrace{\tau_1\frac{\partial^{2} S_{1}}{\partial t^{2}}\bigg|_{P}^{k} + \left[1 - \tau_1	\frac{d F_{1}}{d S_{1}}\bigg|_{P}^{k}\right]\frac{\partial S_1}{\partial t}\bigg|_P^{k} - D_1\frac{\partial^2S_1}{\partial x^2}\bigg|_P^{k} - F_1\bigg|_P^{k}}_{\mbox{PDE}}
\nonumber\\
= \underbrace{\frac{\Delta t}{2}\left(1 - \tau_1\frac{d F_{1}}{d S_{1}}\bigg|_{P}^{k}\right)\frac{\partial^2S_1}{\partial t^2}\bigg|_P^{k}
  -  \frac{\Delta t^2}{3!}\left(1 - \tau_1\frac{d F_{1}}{d S_{1}}\bigg|_{P}^{k}\right)\frac{\partial^3S_1}{\partial t^3}\bigg|_P^{k}}_{\mbox{Local Truncation Error}}
  \nonumber\\
+ \underbrace{ \left(1  -\tau_1\frac{d F_{1}}{d S_{1}}\bigg|_{P}^{k}\right)\mathcal{O}(\Delta t^{3})
 -   2\tau_1\mathcal{O}(\Delta t^{2}) + 2D_1\mathcal{O}(\Delta x^{2})}_{\mbox{Local Truncation Error}}
 \label{vv1_1}
\end{eqnarray}
and
\begin{eqnarray}
\underbrace{\tau_2\frac{\partial^{2} S_{2}}{\partial t^{2}}\bigg|_{P}^{k} + \left[1 - \tau_2	\frac{d F_{2}}{d S_{2}}\bigg|_{P}^{k}\right]\frac{\partial S_2}{\partial t}\bigg|_P^{k} - D_2\frac{\partial^2S_2}{\partial x^2}\bigg|_P^{k} - F_2\bigg|_P^{k}}_{\mbox{PDE}}
\nonumber\\
 = \underbrace{\frac{\Delta t}{2}\left(1 - \tau_2\frac{d F_{2}}{d S_{2}}\bigg|_{P}^{k}\right)\frac{\partial^2S_2}{\partial t^2}\bigg|_P^{k}
  -  \frac{\Delta t^2}{3!}\left(1 - \tau_2\frac{d F_{2}}{d S_{2}}\bigg|_{P}^{k}\right)\frac{\partial^3S_2}{\partial t^3}\bigg|_P^{k}}_{\mbox{Local Truncation Error}}
  \nonumber\\
+ \underbrace{ \left(1  -\tau_2\frac{d F_{2}}{d S_{2}}\bigg|_{P}^{k}\right)\mathcal{O}(\Delta t^{3})
 -   2\tau_2\mathcal{O}(\Delta t^{2}) + 2D_2\mathcal{O}(\Delta x^{2}).}_{\mbox{Local Truncation Error}}
 \label{vv1_2}
\end{eqnarray}
Finally, taking the limit of the continuum, when $\Delta t,\Delta x\to 0$, the equations \ref{vv1_1} and \ref{vv1_2} tend to
\begin{equation}
\label{consistentelimite1}
    \tau_1\frac{\partial^{2} S_{1}}{\partial t^{2}}\bigg|_{P}^{k} + \left[1 - \tau_1	\frac{d F_{1}}{d S_{1}}\bigg|_{P}^{k}\right]\frac{\partial S_1}{\partial t}\bigg|_P^{k} - D_1\frac{\partial^2S_1}{\partial x^2}\bigg|_P^{k} - F_1\bigg|_P^{k} = 0
\end{equation}
and
\begin{equation}
\label{consistentelimite2}
    \tau_2\frac{\partial^{2} S_{2}}{\partial t^{2}}\bigg|_{P}^{k} + \left[1 - \tau_2	\frac{d F_{2}}{d S_{2}}\bigg|_{P}^{k}\right]\frac{\partial S_2}{\partial t}\bigg|_P^{k} - D_2\frac{\partial^2S_2}{\partial x^2}\bigg|_P^{k} - F_2\bigg|_P^{k} = 0.
\end{equation}
Note that the equations (\ref{consistentelimite1}) and  (\ref{consistentelimite2}) are the PDEs (11) and (12), calculated at the mesh point $P$ and at the instant $k$. It is concluded that the discretized equations (23) and (29), or equivalently equations \ref{final1} and \ref{final2}, are consistent with the PDEs (11) and (12), respectively.

\section*{Numerical model stability}
The Von Neumann stability condition is based on the superposition principle, that is, which the error is the superposition of the errors accumulated at each iteration \cite{vneumann1}. The Von Neumann stability condition produces a necessary but not sufficient condition for stability \cite{cuminato1999discretizaccao}.

Consider $\Delta x$ and $\Delta t$ the partitions in space and time, respectively, and $I = \sqrt{-1}$ the imaginary complex number. Let $E_i$, for $i = 1, ... N$, be the  error at each mesh point in time step $t = 0$. Then we write $E_i$ through a complex Fourier series \cite{de2000tecnicas}
\begin{equation}
E_i = \sum_{n=1}^{N}a_{n}^{0} \;\; e^{I\alpha_{n}
(i\Delta x)}
\label{vn88}, \hspace{7mm} \hbox{i = 1, ... N,}
\end{equation}
where $\alpha_{n} = \frac{n\pi}{L}$ and $N\Delta x = L$.

The time dependence of the error $E_i$ is incorporated through the amplitude $a_n^0 \rightarrow a_n^k$. It is assumed that the error tends to grow or decay exponentially with time, so the error \ref{vn88} in time $k$ is written as
\begin{equation}
    E_i^k = \sum_{n=0}^{N}e^{\gamma_n k}e^{I\alpha_{n}i\Delta x}.\label{vn99}
\end{equation}
Now $E_i^k$ is the error at the position $i$ and at the time step $k$.

The error equation \ref{vn99} is a propagation of harmonic waves of the type $e^{\gamma k}e^{I\xi i}$. In order for this propagation to be stable, the absolute value of the error amplitude must be less than or equal to unity \cite{cuminato1999discretizaccao, de2000tecnicas}, i.e.,
\begin{eqnarray}
 |e^{\gamma k}| \leq 1.\label{pratico}
 \end{eqnarray}

Next, the Von Neumann stability conditions will be obtained for the equations \ref{EQ:R_T_C} - \ref{EQ:R_T_C_1} when:

\vspace{0,2cm}
\noindent
Case I) \hspace{0,3cm} $F_1 , F_2 \neq 0$ and $\tau_{1} , \tau_{2} = 0$ (Predator-Prey system with reactive terms).

\vspace{0,2cm}
\noindent
Case II) \hspace{0,3cm} $F_1 , F_2 = 0$, $\tau_{1} , \tau_{2} \neq 0$  (Telegraph equations).

\vspace{0,2cm}
\noindent
Case III) \hspace{0,3cm} $F_1 , F_2 \neq 0$ and $\tau_{1} ,  \tau_{2}\neq 0$ (Telegraph Predator-Prey system). In this case the stability analysis will be obtained through numerical experiments.

\subsection*{Von Neumann conditions for a Predator-Prey system}\label{sectioncalor2}
Consider the following Diffusive Predator-Prey system
\begin{eqnarray}
	\frac{\partial S_{1}}{\partial t}  = D_1\frac{\partial^2 S_{1}}{\partial x^2} + a_1S_1 - b_1S_1^2 - c_1S_1S_2
	\label{calor1}
	\\
	\frac{\partial S_{2}}{\partial t}  = D_2\frac{\partial^2 S_{2}}{\partial x^2} - a_2S_2 +  c_2S_1S_2,\;\;\;\;\;\;
	\label{calor2}
\end{eqnarray}
where $S_1$, $S_2$ are the densities of prey and predator populations, respectively. The parameters $a_1$, $a_2$, $c_1$, $c_2$, and $b_1$ are positive constants described in (15) and (16), while $D_1>0$ and $D_2>0$ are the prey and predator diffusion rates, respectively.  Note that the equations \ref{calor1} and \ref{calor2} are the equations \ref{EQ:R_T_C} and \ref{EQ:R_T_C_1} when $\tau_1 = \tau_2 = 0$.

For the discretization of the equations \ref{calor1} and \ref{calor2}, we use progressive finite differences in time first-order derivative and central finite differences in space second-order derivative. Then the discretized versions of the equations \ref{calor1} and \ref{calor2} have the form
\begin{eqnarray}
S_1|_P^{k+1} = S_1|_{P}^{k} + \sigma_1\left(S_1|_E^{k} - 2S_1|_P^{k} + S_1|_W^{k}\right) \nonumber \\
+ \Delta tS_1|_P^{k}\left(a_1 - b_1S_1|_{P}^{k} - c_1S_2|_{P}^{k}\right) \label{calor1dis}
\\
S_2|_P^{k+1} = S_2|_{P}^{k} + \sigma_2\left(S_2|_E^{k} - 2S_2|_P^{k} + S_2|_W^{k}\right) \nonumber \\
+ \Delta tS_2|_P^{k}\left(- a_2 +  c_2S_1|_{P}^{k}\right)
\;\;\;\;\;\;\;\;\;\;\;\;\;
\label{calor2dis}
\end{eqnarray}
where $\sigma_j =\frac{\Delta t \; D_j}{\Delta x^2}$ for $j = 1,2$ .

Note that the equations \ref{calor1dis} and \ref{calor2dis} are non-linear, due to the reactive terms $F_1$ and $F_2$. On the other hand, the Von Neumann superposition procedure is valid only for linear equation systems. Then it is necessary to linearize the equations \ref{calor1dis} and \ref{calor2dis}.

It is assumed, in the nonlinear terms of \ref{calor1dis} and \ref{calor2dis}, that the variables $S_1$ and $S_2$ are locally positive constants $m_1$ and $m_2$, respectively. From this hypothesis, it follows that the linear version of the equations \ref{calor1dis} and \ref{calor2dis} is given by
\begin{eqnarray}
S_1|_P^{k+1} = S_1|_{P}^{k} + \sigma_1\left(S_1|_E^{k} - 2S_1|_P^{k} + S_1|_W^{k}\right) \nonumber \\
+ \Delta tS_1|_P^{k}\left(a_1 - b_1m_1 - c_1m_2\right)
\label{calor1dislinear}
\\
S_2|_P^{k+1} = S_2|_{P}^{k} + \sigma_2\left(S_2|_E^{k} - 2S_2|_P^{k} + S_2|_W^{k}\right) \nonumber \\
+ \Delta tS_2|_P^{k}\left(- a_2 +  c_2m_1\right).
\;\;\;\;\;\;\;\;\;
\label{calor2dislinear}
\end{eqnarray}

Assuming that $\left.S_{1} \right|_{P}^{k} = \left.S_{2} \right|_{P}^{k} = e^{\gamma k}e^ {I\xi P}$ in \ref{calor1dislinear} and \ref{calor2dislinear}, we obtain for the point $P$, in the time $k$,
\begin{eqnarray}
e^{\gamma} = 1 - 4\sigma_1\sin^2 \frac{\xi}{2} + \Delta t\left(a_1 - b_1m_1 - c_1m_2\right)\label{vncalor1} \\
e^{\gamma} = 1 - 4\sigma_2\sin^2 \frac{\xi}{2} + \Delta t\left( - a_2 + c_2m_1\right). \;\;\;\;\;
\label{vncalor2}
\end{eqnarray}
The stability condition requires that $|e^{\gamma}| \leq 1$, so
\begin{eqnarray}
- 1 \leq 1 - 4\sigma_1\sin^2 \frac{\xi}{2} + \Delta t\left(a_1 - b_1m_1 - c_1m_2\right) \leq 1\label{11} \\
- 1 \leq  1 - 4\sigma_2\sin^2 \frac{\xi}{2} + \Delta t\left( - a_2 + c_2m_1\right) \leq 1.\label{12} \;\;\;\;
\end{eqnarray}
Finally, after some algebraic manipulations, we get
\begin{eqnarray}
0 \leq \Delta t \leq \frac{\Delta x^2}{2D_1\sin^2 \frac{\xi}{2} - \frac{\Delta x^2}{2}\left(a_1 - b_1m_1 - c_1m_2\right)}
\label{12a} \\
0 \leq \Delta t \leq \frac{\Delta x^2}{2D_2\sin^2 \frac{\xi}{2} - \frac{\Delta x^2}{2}\left(- a_2 + c_2m_1\right)},
\label{12b} \;\;\;\;
\end{eqnarray}
which are the Von Neumann conditions for the predator-prey discretized equations \ref{calor1dis} and \ref{calor1dis}, respectively. Note that the conditions \ref{12a} and \ref{12b} impose restrictions on the spatial refinements $\Delta x$ and temporal refinements $\Delta t$ for stability in the numerical experiments.

\subsection*{Von Neumann conditions for a Telegraph system}\label{sectionsera}
Consider the telegraph equation system
\begin{eqnarray}
\tau_{j}\frac{\partial^2 S_{j}}{\partial t^2} + \frac{\partial S_{j}}{\partial t}  = D_{j}\frac{\partial^2 S_{j}}{\partial x^2},\label{Von1}
\end{eqnarray}
where $\tau_j>0$ and $D_j>0$ are delay and diffusion constants, respectively, for $j = 1,2$.

Using the discretization schemes given in \ref{eqsjtt}-\ref{eqsjt} in \ref{Von1}, it follows that
\begin{eqnarray}
\left.S_{j} \right|_{P}^{k+1} = \left(2 - \frac{\Delta t}{\tau_j} - \frac{2D_j{\Delta t}^2}{{\Delta x}^2\tau_j}\right)\left.S_{j} \right|_{P}^{k} \;\;\;\;\;\;\;\;\;\;\; \nonumber \\
+ \frac{D_j{\Delta t}^2}{\Delta x^2\tau_j}\left(\left.S_{j} \right|_{E}^{k} + \left.S_{j} \right|_{W}^{k}\right) - \left(1 - \frac{\Delta t}{\tau_j}\right)\left.S_{j} \right|_{P}^{k-1}.\hspace{1mm}\label{Vondis}
\end{eqnarray}
From the von Neumann hypothesis $\left.S_{j} \right|_{P}^{k} = e^{\gamma k}e^{I\xi P}$, we get that
\begin{eqnarray}
e^{\gamma} = 2 - \frac{\Delta t}{\tau_j} - \frac{2D_j\Delta t^2}{\tau_j\Delta x^2}\left(1 - \cos{\xi} \right) - \left(1 - \frac{\Delta t}{\tau_j}\right)e^{-\gamma}.\label{Von1_1}
\end{eqnarray}
Multiplying \ref{Von1_1} by $e^{\gamma}$ and making use of trigonometric transformations, equation \ref{Von1_1} is rewritten as
\begin{eqnarray}
e^{2\gamma} - 2\beta_j e^{\gamma} + \left(1 - \frac{\Delta t}{\tau_j}\right) = 0,\label{Von1_2}
\end{eqnarray}
where $\beta_j = 1 - 2\sigma_j\frac{\Delta t}{\tau_j}\left(\sin^2{\frac{\xi}{2}}\right) - \frac{\Delta t}{2\tau_j}$ and $\sigma_j =  \frac{D_j\Delta t}{\Delta x^2}$, for $j = 1, 2$. Solving the roots of \ref{Von1_2}
\begin{equation}
    g_{1,2} = e^{\gamma} = \beta_j \pm \sqrt{\beta_j^2 - 1 + \frac{\Delta t}{\tau_j}}.\label{nutela}
\end{equation}

To analyze the roots \ref{nutela}, consider the following lemma.

\vspace{0.5cm}
\noindent
LEMMA I: Let $x \in (0.1]$ and $a(x)$ be a real function defined by $a(x) = \frac{x}{2}$. If for each $x \in (0.1 ]$ we take $y \leq 1 - a(x)$, then
  $0 \leq \sqrt{y^2 - 1 + x} + y \leq 1$.
\vspace{0.5cm}

\noindent
Proof: Let $y$ be a real number such that $y \leq 1 -\frac{x}{2}$ for all $x \in (0,1]$. Thus, we have $y \in \left[\frac{1}{2},1\right)$. So,
  \begin{eqnarray*}
  0 \leq\sqrt{y^2 - 1 + x} + y & \leq & \sqrt{ \left(1 - \frac{x}{2}\right)^2 - 1 +x} + 1 -\frac{x}{2} \\
  & = & \sqrt{\frac{x^2}{4}} + 1 - \frac{x}{2}=1.
  \end{eqnarray*}
Therefore, $0 \leq \sqrt{y^2 - 1 + x} + y \leq 1$.
 $\hfill\Box$
\vspace{0,5cm}

In the context of LEMMA I, consider the following cases:

\vspace{0,5cm}
\noindent
Case I) \hspace{0,3cm} $\beta_j^2-1 + \frac{\Delta t}{\tau_j} < 0$.

\noindent
In this case $\sqrt{\beta_j^2 - 1 + \frac{\Delta t}{\tau_j}}$ generates conjugate complex roots, that is,
 \begin{equation}
  g_{1,2} = \beta_j \pm i\sqrt{1 - \frac{\Delta t}{\tau_j} - \beta_j^2}\; .\label{ven}
 \end{equation}
 Taking the norm of\ref{ven}
 \begin{equation}
    ||g||^2 = \beta_j^2 + 1 - \frac{\Delta t}{\tau_j} -\beta_j^2 = 1 - \frac{\Delta t}{\tau_j} < 1
    \hspace{0,05cm}, \mbox{if} \hspace{0,2cm} 0 \leq \frac{\Delta t}{\tau_j} \leq 1.
  \end{equation}
 So, in this case, when  $0 \leq \frac{\Delta t}{\tau_j} \leq 1$ and $\beta_j^2 - 1 + \frac{\Delta t}{\tau_j} < 0$, by Von Neumann's criterion there is numerical convergence for the \ref{Vondis} system.

\vspace{0,5cm}
\noindent
Case II) \hspace{0,3cm} $\beta_j^2-1 + \frac{\Delta t}{\tau_j} \geq 0$.

\noindent
In this case, for $0 \leq \frac{\Delta t}{\tau_j} \leq 1$, equation (62) can result in $g_{j} = e^{\gamma} = 2\beta_j$, when $\frac{\Delta t}{\tau_j}=1$, therefore we must limit the possible values of $\beta_j$. From LEMMA I, we impose that
\begin{equation}
\frac{1}{2} \leq \beta_j\leq 1 -
\frac{\Delta t}{2\tau_j},
\end{equation}
which results in
\begin{equation}
\frac{1}{2} \leq  1 - 2\sigma_j\frac{\Delta t}{\tau_j}\left(\sin^2{\frac{\xi}{2}}\right) \leq 1.
\end{equation}
Solving the inequality (66), it follows that
\begin{eqnarray}
0 \leq \sigma_j \leq \frac{1}{4}.
\label{condicaoSigma1}
\end{eqnarray}
Then, by LEMMA I, if $0 < \frac{\Delta t}{\tau_j} \leq 1$ and $\beta_j^2-1 + \frac{\Delta t}{\tau_j} \geq 0$, the Von Neumann stability condition of the discretized equation \ref{Vondis} is given by
\begin{eqnarray}
0 \leq \sigma_j \leq \frac{1}{4},
\label{condicaoSigma11}
\end{eqnarray}
where $\sigma_j =  \frac{D_j\Delta t}{\Delta x^2}$ for $j = 1, 2$.

\subsection*{Stability Diagram for a Telegraph Predator-Prey System}\label{secaovixe}
In the case of equations \ref{EQ:R_T_C} and \ref{EQ:R_T_C_1}, or in its discretized form given by the equations \ref{finalpresa} and \ref{finalpredador}, which describe a Telegraph Predator-Prey system with diffusive, reactive, and delayed effects, it was not possible to obtain an analytical form for the Von Neumann stability condition. In this context, numerical experiments will be carried out to obtain the stability diagram for the numerical scheme used.

Setting the reactive parameters $a_1 = 1 \;s^{-1}$, $a_2 = 0.75 \;s^{-1}$, $b_1 = 0.5 \; m/s$, $c_1 = 0.5 \; m/s$, and $c_2 = 0.5 \; m/s$, phase diagrams of the variables $D_j$ (diffusibility coefficient) and $\tau_j$ (delay time) were constructed, for $j = 1, 2$. The phase diagrams show the regions of stability and instability of the our PDEs system, for different discretizations of $\Delta t$ and $\Delta x$. It is important to note that $D_1 = D_2$ and $\tau_1 = \tau_2$ in all simulations, that is, there was no different diffusion or delay between the populations $S_1$ and $S_2$. In the following numerical experiments, the following initial and boundary conditions were considered
 \begin{eqnarray}
      S_{1}(x,0) = S_{1}^{0}=S_{2}(x,0) = S_{2}^{0}=
      \begin{cases}
15 &\mbox{if} \hspace{0.4cm} 24 \leq x \leq 26\\
0 &\mbox{if} \hspace{0.4cm} \mbox{otherwise} \\
      \end{cases}
\label{condicaoinicial1_1}\\ \nonumber  \\
S_{1}(0,t)=S_{1}(50,t)=S_{2}(0,t)=S_{2}(50,t) = 0,
\;\;\;\;\;\;\;\;\;
\label{condicaodecontorno_1}
       \end{eqnarray}
where $x\in[0,50]$ and $t\in[0,100]$.

All the simulations contained in this work were carried out in an Ubuntu operating system, where the codes were all programmed in FORTRAN 90 language and the images generated via gnuplot software.

First, some simulations were carried out with different partitions of the temporal domain. Fixed $\Delta x = 0.1 $, Figures 3 and 4 show the graphs of the numerical stability/instability regions, as a function of the diffusion ($D_j$) and the delay time ($\tau_j$) of prey and predator populations, for two different values of $\Delta t$, $\Delta t=0.002$ in Figure 3 and $\Delta t=0.001$ in Figure 4.
\begin{figure}[H]
	\begin{center}
	\caption{Region of stability/instability for the Telegraph Predator-Prey system for refinements $\Delta x = 0.1 $ and $\Delta t = 0.002$.}
\includegraphics[width=9.5cm]{usar3.png}
\label{lol11}
	\end{center}

\end{figure}
\begin{figure}[H]
	\begin{center}
	\caption{Region of stability/instability for the Telegraph Predator-Prey system for refinements $\Delta x = 0.1 $ and $\Delta t = 0.001$.}
\includegraphics[width=9.5cm]{usar1.png}
\label{lol22}
	\end{center}
\end{figure}

Note that when refining the temporal domain, to a fixed $\Delta x$, the numerical stability region grows. See the values of $D_j$ in Figures 3 and 4.

Now we set $\Delta t = 0.0015255$. Figures 5 and 6 show the graphs of the numerical stability/instability regions, as a function of the diffusion ($D_j$) and the delay time ($\tau_j$) of prey and predator populations, for two different values of $\Delta x$, $\Delta x = 0.1$ in Figure 5 and $\Delta x=0.025$ in Figure 6.

\begin{figure}[H]
	\begin{center}
	\caption{Region of stability/instability for the Telegraph Predator-Prey system for refinements $\Delta x = 0.1 $ and $\Delta t = 0.0015255$.}
\includegraphics[width=10.0cm]{usar4.png}
\label{lol1}
	\end{center}
\end{figure}
\begin{figure}[H]
	\begin{center}
	\caption{Region of stability/instability for the Telegraph Predator-Prey system for refinements $\Delta x = 0.025$ and $\Delta t = 0.0015255$.}
\includegraphics[width=10.0cm]{usar6.png}
\label{lol123}
	\end{center}
\end{figure}

Note that when refining the spatial domain, to a fixed $\Delta t$, the numerical stability region decreases. See the values of $D_j$ in Figures 5 and 6.

\section*{Numerical Convergence Analysis}
The objective of this section is to analyze the convergence of the numerical scheme used to solve the model under study. In general, to verify numerical convergence, the Lax Equivalence Theorem must be satisfied \cite{lalax,laxlutor,romeiro1}.

\vspace{0.3cm}
\noindent
Lax Equivalence Theorem:
For a consistent finite difference scheme of a well-posed initial value problem, stability is a necessary and sufficient condition for convergence.
\vspace{0.0cm}

First, it is necessary to verify if the PDEs system are well-posed, that is, if the problem in question has a solution and if it is unique. In the literature there are works on the existence and uniqueness of solutions of Telegraph equations \cite{telegraf}. There are also several works on the existence and uniqueness of Telegraph Predator-Prey systems, or more general Telegraph models  \cite{todabem1,toda,toda1,todabem,toda2}.

Then, a numerical study of mesh refinement is presented to evaluate the process of numerical convergence.

\subsection*{Mesh Refinement and Convergence}
Note that $S_{1}$ and $S_{2}$ are population densities of prey and predators, respectively, so to obtain the total population of prey and predators, in a time $t$, one must integrate $ S_{1}$ and $S_{2}$ at the position coordinate $x$. This calculation is performed using the definite integral
\begin{equation}
P_{j}(t) = \int\limits_{X_{INI}}^{X_{FIN}} S_{j}(x,t)dx,
\end{equation}
where $P_{1}$ and $P_{2}$ are the populations of prey and predator, respectively, at time $t$.

Let $NI$ be the number of partitions in the spatial domain and $NJ$ the number of partitions in the temporal domain. Thus, we have the following relations
\begin{eqnarray}
\Delta t = \frac{T_{FIN} - T_{INI}}{NJ - 1}\label{dt}
\;\;\;\; \mbox{and} \;\;\;\;
\Delta x = \frac{X_{FIN} - X_{INI}}{NI - 1}.\label{dx}
\end{eqnarray}
Next, an analysis of the convergence of the numerical solution of the problem regarding the mesh refinement is performed. The Table \ref{tabelatestetau0} presents the values of the parameters used in the simulations in this subsection. The initial and boundary conditions are those given earlier in \ref{condicaoinicial1_1} and \ref{condicaodecontorno_1}.
\begin{table}[H]
\renewcommand{\arraystretch}{1.2}
\centering
\caption{Values for the model parameters used in the numerical simulations presented in Table 2.}
\label{tabelatestetau0}
\begin{tabular}{ccc}
\hline
\textbf{Parameters}                                      & \textbf{Prey } & \textbf{Predator } \\ \hline
\textbf{$a_{j} \; (s^{-1})$}       & 1.0                    & 0.75                        \\
\textbf{$b_{j} \; (m/s)$}       & 0.5                    & 0.0                        \\
\textbf{$c_{j} \; (m/s)$}        & 0.5                    & 0.5                        \\
\textbf{$D_{j} \; (m^2/s)$}         & 1.0                    & 1.0                        \\
\textbf{$\tau_{j} \; (s)$}         & 0.001                    & 0.001                        \\
$ T_{INI} \; (s)$              & 0.0                  & 0.0                      \\
$ T_{FIN} \; (s)$              & 100.0                  & 100.0                      \\
$ X_{INI} \; (m)$              & 0.0                  & 0.0                      \\
$ X_{FIN} \; (m)$              & 50.0                  & 50.0 \\ \hline
\end{tabular} \\
\end{table}

Table \ref{tabelatudo} presents numerical experiments for the solution of the discretized Telegraph Predator-Prey system (23-32), as a function of the mesh refinement. The process of population convergence is observed as a function of this refinement.
\begin{table}[H]
\renewcommand{\arraystretch}{1.2}
\centering
\caption{Prey population $P_{1}(t) $ and predator population $P_{2}(t) $  of the Telegraph Predator-Prey model for various refinements $\Delta x$ and $\Delta t$, at time $t = 100 \; s$, with the parameters in Table 1.}
\label{tabelatudo}
\begin{tabular}{cccc}
\hline
\textbf{$\Delta x$ $(m)$}                                      & \textbf{$\Delta t$ $(s)$ } & \textbf{$P_{1}(t=100) $}   & \textbf{$P_{2}(t=100) $}\\ \hline
5.0    & 0.002               & 62.69325     & 18.90470
\\
3.5    & 0.002               & 65.13776     & 19.45726
\\
2.0    & 0.002               & 70.27077     & 20.51610
\\
1.0    & 0.002               & 71.68832     & 21.04255
\\
0.75    & 0.00175               & 72.22790     & 21.17758
\\
0.5    & 0.0015               & 72.56129    & 21.29889
\\
0.1    & 0.001               & 73.13264    & 21.49681
\\
 0.05       & 0.0005              & 73.20856   & 21.52185
\\
0.04        & 0.0004              & 73.22319  & 21.52683
\\
0.03         & 0.0003          & 73.23832       & 21.53186
\\
0.02       & 0.0002              & 73.25332    & 21.53683                        \\
0.01        & 0.0001              & 73.26829    & 21.54183                        \\
0.005        & 0.00005              & 73.27579    & 21.54433                        \\
0.0025        & 0.000025              & 73.27954    & 21.54558                        \\ \hline
\end{tabular}\\
\end{table}

\subsection*{Numerical Experiments on Convergence in the Telegraph Predator-Prey System}\label{simu1}
In this subsection, two numerical experiments are carried out
for the Telegraph Predator-Prey system (11-12) in order to better understand the stability/instability phase diagrams obtained in Figures 3-6.

Let us consider the stability and instability regions obtained in Figure 5, when $\Delta t=0.0015255$ and $\Delta x=0.1$. Figure 7 presents these regions again. Figure 7 also shows two points, one in the region of stability of the numerical scheme and the other at the limit of the region of instability of the numerical scheme.
\begin{figure}[h]
    \begin{center}
    \caption{Region of stability/instability for the Telegraph Predator-Prey system for refinements $\Delta x = 0.1$ and $\Delta t = 0.0015255$. The point $ (0.05,20)$ is in the stability region, while the point $ (0.05,61)$ is at the limit
of the instability region.}
\includegraphics[width=10.0cm]{regiao.png}
\label{oae}
	\end{center}
\end{figure}

For the first numerical experimentation in this subsection, the values of the model parameters are presented in Table \ref{tabelatestetau111}. Note that this simulation corresponds to the point in the stability region of Figure 7.
\begin{table}[H]
\renewcommand{\arraystretch}{1.2}
\centering
\caption{Model parameter values for the first numerical experiment, corresponding to the point in the stability region of Figure 7.}
\label{tabelatestetau111}
\begin{tabular}{ccc}
\hline
\textbf{Parameters}                                      & \textbf{Prey }  & \textbf{Predator }  \\ \hline
\textbf{$a_{j} \; (s^{-1})$}       & 1.0                    & 0.75                        \\
\textbf{$b_{j} \; (m/s)$}       & 0.5                    & 0.0                        \\
\textbf{$c_{j} \; (m/s)$}        & 0.5                    & 0.5                        \\
\textbf{$D_{j} \; (m^2/s)$}         & 20.0                    & 20.0                        \\
\textbf{$\tau_{j} \; (s)$}        & 0.05                    & 0.05                        \\ \hline
\end{tabular} \\
\end{table}

Figure \ref{log2a} presents the population densities of prey and predators, in space coordinate, for the first numerical experiment at time $t=100$.
\begin{figure}[ht]
\begin{center}
\caption{Population densities of prey and predators of the Telegraph Predator-Prey model, in $t=100$, corresponding to the first numerical experiment.}
\includegraphics[width=10.0cm]{dencerto1.png}
\label{log2a}
\end{center}
\end{figure}

Figure \ref{log2b} presents the temporal variation, in $x=25$, of the populations of prey and predators in the first numerical experiment.

Finally, Figure \ref{log22} shows population densities of predator and prey in the entire spatial and temporal domain. Note that in this simulation there are no negative population densities.
\begin{figure}[H]
\begin{center}
\caption{Populations of prey and predators over time of the Telegraph Predator-Prey model, in $x=25$, corresponding to the first numerical experiment.}
\includegraphics[width=9.5cm]{popcerto1.png}
\label{log2b}
\end{center}
\end{figure}

\begin{figure}[H]
\begin{center}
\caption{3D graph of predator and prey population densities of the Telegraph Predator-Prey model corresponding to the first numerical experiment.}
\includegraphics[width=11.0cm]{vaicerto1.png}
\label{log22}
\end{center}
\end{figure}

For the second numerical experiment in this subsection, the values of the model parameters are presented in Table \ref{tabelatestetau233}. The second numerical experiment corresponds to the point at the limit of the instability region in Figure 7.
\begin{table}[H]
\renewcommand{\arraystretch}{1.2}
\centering
\caption{Model parameter values for the second numerical experiment, corresponding to the point at the limit of the instability region in Figure 7.}
\label{tabelatestetau233}
\begin{tabular}{ccc}
\hline
\textbf{Parameters}                                      & \textbf{Prey } & \textbf{Predator } \\ \hline
\textbf{$a_{i} \; (s^{-1})$}       & 1.0                    & 0.75                        \\
\textbf{$b_{i} \; (m/s)$}       & 0.5                    & 0.0                        \\
\textbf{$c_{i} \; (m/s)$}        & 0.5                    & 0.5                        \\
\textbf{$D_{i} \; (m^2/s)$}         & 61.0                    & 61.0                        \\
\textbf{$\tau_{i} \; (s)$}        & 0.05                    & 0.05                        \\ \hline
\end{tabular} \\
\end{table}

Figure \ref{fig:gull} presents the population densities of prey and predators, in space coordinate, for the second numerical experiment at time $t = 100$. Note that the final population density of predators is zero, that is, the extinction of predators occurs.
\begin{figure}[ht]
\caption{Population densities of prey and predators of the Telegraph Predator-Prey model, in $t=100$, corresponding to the second numerical experiment.}
\begin{center}
\includegraphics[width=10.0cm]{denerrado1.png}
\label{fig:gull}
\end{center}
\end{figure}

Figure \ref{fig:tiger} presents the temporal variation, in x = 25, of prey and predators populations in the second  numerical experiment.
\begin{figure}[H]
\begin{center}
\caption{Populations of prey and predators over time of
the Telegraph Predator-Prey model, in x = 25 , correspond-
ing to the second numerical experiment.}
\includegraphics[width=10.0cm]{manodo.png}
\label{fig:tiger}
\end{center}
\end{figure}

The Figure \ref{fig:tiger2} shows population densities of predator and prey in the entire spatial and temporal domain. It is observed that for some values of $t$ there are negative population densities, as can be seen in Figure \ref{zoom}.
\begin{figure}[H]
\begin{center}
\caption{3D graph of predator and prey population densities of the Telegraph Predator-Prey model corresponding to the second numerical experiment.}
\includegraphics[width=11.2cm]{vaierrado1.png}
\label{fig:tiger2}
\end{center}
\end{figure}

\begin{figure}[ht]
\begin{center}
\caption{Observation (first red line) of the spatial/temporal region where negative solutions for prey population density occur.}
\includegraphics[width=10.0cm]{zoom1.png}
\label{zoom}
\end{center}
\end{figure}

Note this behavior is unrealistic from a biological point of view. According to the numerical solution presented in the second experiment, given the initial condition (69), the prey population rapidly decreases, becomes negative and grows again until it reaches saturation. This is evidence of the instability of numerical solutions.

\section*{Conclusion}
In this work, the mathematical modeling of the   diffusive-reactive predator-prey system with delay (11)-(12) was performed. This model is also called Telegraph Predator-Prey system. Through numerical modeling, this EDP system was discretized by the Finite Difference method, equations (23)-(32).

First, it was verified that the discretized equations (23)-(32) were consistent with the PDEs (11)-(12) of the mathematical model.

In the sequence, through the Von Neumann procedure, the numerical stability of systems simpler than the Telegraph Predator-Prey system was discussed. For the predator-prey system (46)-(47) and for the telegraph equation system (58) it was found that the numerical scheme used is conditionally stable, since the stability constraints (56)-(57) and (68) depend on the parameters of these models.

On the other hand, for the telegraph predator-prey system (11)-(12), it was not possible to obtain an analytical mathematical relationship for the Von Neumann stability condition. Then, through numerical experiments, phase diagrams of the stability/instability regions were constructed as a function of the diffusion $D_j$ and delay time $\tau_j$ parameters of the model (11)-(12), see Figures 3-6. Thus, it can be concluded that the telegraph predator-prey system (11)-(12) is also conditionally stable, depending on its diffusion, delay and reactive parameters.

Through numerical experiments again, the convergence of the numerical model was analyzed. Through the spatial and temporal refinement of the mesh, it was found that the prey and predator populations of the telegraph predator-prey system converged, as shown in Table 2.

Finally, it was possible to observe the numerical instabilities that occur in numerical simulations. In the phase diagram presented in Figure 7, two points were chosen, one in the region of stability of the numerical scheme and the other at the limit of the region of instability of the numerical scheme. It was found that when passing from the stability region to the instability region, negative fluctuations occur for population densities, which is an evidence of instability of numerical solutions, as can be seen in Figure 14.

\section*{Acknowledgments}
The authors Kariston Stevan Luiz and Juniormar Organista was partly financed by the Coordination of Improvement of Higher Education Personnel (Capes), Brazil - Finance Code 001.

redpajama_set_name: RedPajamaArXiv


text:
Dustin Johnson Misses Cut at PGA, Snaps Longest Active Streak
By Peter Bukowski
Dustin Johnson was the favorite coming into the week, but he'll leave early, missing the cut at Baltusrol.
Andrew Redington/Getty Images
SPRINGFIELD, N.J. — Dustin Johnson entered the PGA Championship as the favorite. He'll leave Friday finishing as one of the last players in the field.
Johnson shot a brutal seven-over 77 in Round 1, including making an absolute mess of the 18th when he hit his drive into the water and hit a tree with his second, making bogey.
Johnson was marginally better in Round 2, posting a two-over 72 but will still miss the cut, which will snap the longest active streak on Tour at 25 consecutive made cuts.
The No. 2 player in the world had a chance to take over the top spot in the World Golf Ranking depending on how the tournament shook out between himself and world No. 1 Jason Day.
Those hopes are gone, and soon too will be Johnson … at least for this week.
SCORES: View the Leaderboard


Catch the candid interview of Vishwas Mudagal with RJ Darius on Fever 104 FM on the famous show Dons of Bengaluru.
Vishwas shares his journey as an author, as an entrepreneur. He also talks about his best-selling books – Losing My Religion and The Last Avatar. He also shares the growth story of his two successful business ventures GoodWorkLabs and GoodWorks CoWork.


text:
"""
The purpose of this module is to run a single test associated
with a project.
"""

import argparse
import traceback
import os
import subprocess
import sys
import re

# Number of seconds before killing command
TEST_TIMEOUT = 2
PROJECT_EXECUTABLE = "cli.py"

def remove(filename):
    if os.path.exists(filename):
        os.remove(filename)

class TestRunnerException(Exception):
    """
    Exception used to distinguish if the test runner failed.
    """
    pass

def get_output_from_args(args, timeout):
    output = []
    try:
        stdout = subprocess.check_output(args,
                                         stderr=subprocess.STDOUT,
                                         universal_newlines=True,
                                         timeout=timeout)
        output.append(stdout)
    except subprocess.CalledProcessError as cpe:
        output.append("Non-Zero Return Code")
    except subprocess.TimeoutExpired as te:
        output.append("TestRunner: {} took too long, killing it.".format(
            args[0]))
    return output

def get_input(test_file):
    """
    Returns the contents of the test.
    """
    return test_file.read()

if __name__ == "__main__":
    parser = argparse.ArgumentParser(description="""
    Runs a single test.
    If no arguments (apart from the test filename) are given,
    it outputs your project's test output.
    """)
    parser.add_argument('test_file', type=argparse.FileType('r'),
                        help="The file containing the test contents.")
    parser.add_argument('output_file', nargs='?', type=argparse.FileType('w'),
                        default=sys.stdout, help="""The file to write the output of this program to.
If not given, outputs to stdout.""")
    parser.add_argument('--correct', action='store_true', help="""
    If given, outputs the correct output for a test.""")
    parser.add_argument('--input', action='store_true', help="""
If given, outputs the contents of the test.""")

    args = parser.parse_args()
    if args.input:
        output = args.test_file.read()
    elif args.correct:
        chars_to_remove = len("input.txt")
        correct_file_name = args.test_file.name[:-chars_to_remove]
        correct_file_name += "correct.txt"
        output = open(correct_file_name, 'r').read()
    else:
        executable_args = ["python3",
                           PROJECT_EXECUTABLE,
                           args.test_file.name,
                           "output.txt"]
        get_output_from_args(executable_args, TEST_TIMEOUT)
        output = open("output.txt", 'r').read()

    args.output_file.write(output)
    remove("output.txt")

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Do you have a loved one struggling with a drug or alcohol addiction? You're not alone. An estimated 20.2 million American adults have a substance use disorder.
For loved ones, addiction feels frightening and confusing. It can also feel downright infuriating.
Help is available. However, it can be challenging to guide your loved one to the right support and treatment if they have been resistant to your efforts in the past.
Staging an intervention can make a profound difference in supporting your loved one. Let's get into how to choose an intervention specialist.
What Does An Intervention Specialist Do?
In the intervention, family members attempt to convince an addict or alcoholic to seek treatment. Loved ones convene to discuss how the substance use has impacted their lives.
During this conversation, family members usually set boundaries indicating what they will do if the loved one refuses their request. Boundaries may include kicking the addict out of the home or refusing to provide them money.
Intervention specialists have specialized training in addiction. Additionally, they have seasoned experience working with both addicts and their families during this process.
When Do You Hire This Professional?
Some families hire their specialists a few days before staging the intervention. Others may start weeks before the intervention is scheduled to occur.
Most specialists hold meetings with the family and loved ones before the event. This helps them mentally prepare them for the discussion ahead. During this process, some loved ones will back out of the process (change, after all, is scary).
However, interventionists work like coaches. They educate everyone on how to describe their feelings and reactions. They also act as a neutral party that remains emotionally disconnected from the family.
While interventionists typically provide support before and during the intervention, some of them stay in touch with the family during and after treatment.
This helps provide support throughout the process. Interventionists can continue intervening if and when problems arise.
It should be noted that relapse can be a part of the recovery process. A trusted advisor can help you set strong boundaries and take care of yourself during this challenging time.
Before choosing your specialist, you need to consider a variety of factors.
What kind of experience does the specialist have? What have their success rates been in the past? Can they provide references from past clients or families who they worked with in the past?
Ideally, any specialist should have extensive experience with planning, implementing, and staging family interventions.
That said, it's a red flag if a specialist boasts about an exceptionally high success rate. Addiction recovery is a notoriously challenging process. Proceed with caution if anyone guarantees you certain results.
If you belong to a support group (like Al-Anon), feel free to ask your peers if they know of an interventionist they would recommend.
Some people 'freelance' as interventionists, but those individuals might not be the most qualified for the job. You should always inquire about education. While requirements vary, specialists often have psychology or social work degrees.
These careers require advanced degrees, which can indicate increased addiction knowledge and expertise.
You can also check out the Association of Intervention Specialists (AIS). They refer to a network of interventionists throughout the country.
Some insurances will cover the costs of intervention treatments. However, this is often an exception and not a rule. Usually, families must fork over some cash. Always check with your healthcare provider to determine what services will (or will not) be covered.
Make sure to inquire about the price of services and what exactly those include upon hiring. You can also check to see if specialists offer a sliding scale or financing options.
Good interventionists have a professional understanding of addiction. With that, they also have strong boundaries, communication skills, compassion, and empathy.
It's a red flag if they seem overly fixated or rigid on a particular modality of treatment or treatment program. After all, addiction is a multi-faceted disease, and professionals recognize that all people have different needs when it comes to what works best for them.
Finally, a good interventionist always puts the client's safety as a top concern- even if it means putting it in front of the family or friends' wants. If someone needs medical stabilization, that should always be at the forefront of the priority list.
At the end of the day, most families choose the professional based on how they feel. In other words, do you trust this person? Do you feel like they are a good fit for your family? Do you like how they articulate themselves?
Degrees, experience, and certification matter, but it's essential that you listen to your intuition during this process. After all, you're going to be spending time with this person discussing some of your most intimate fears. It's critical that you feel safe and supported!
Witnessing someone you love struggle with addiction can be one of the most painful and distressing experiences someone can endure.
You can, however, make a lasting impression. While interventions may seem harsh and heartless, they may be the necessary action your loved one needs to recognize the problem at hand. Hiring the right intervention specialist can make this process easier for everyone involved.
Interested in finding optimal, professional addiction support? Check out our services today!


Monday regular hours are 8- 6 PM after this time one message will play till 10PM and then after 10 PM there will be another message (regular after hour message. This message will be playing for the rest of the week. So the only problem day is Monday between 6 PM and 10 PM. All other days are 8-6 regular hours.
Looks like the service flag is the way to go.
You'll need 2 service flags. Under the Night service setting you'll set Service flag 2 will 1st then set the 2nd Service flag under the same account.