text:
Home Front Command drill (archives(
Photo: Roee Idan
'Cooperation to benefit Europe and Israel.' Minister Vilnai
Photo: Amit Shabi, Yedioth Ahronoth
NATO forces to attend drill in Israel
Organization's search and rescue units, emergency services to take part for first time in Home Front Command drill simulating massive earthquake
Yoav Zitun|Published: 01.02.12 , 01:06
NATO forces are scheduled to participate, for the first time, in a home front drill that will be held in Israel next October.
In recent years, the IDF took part in several exercises held by the North Atlantic Treaty Organization abroad, but has never hosted its members in Israel.
The drill, dubbed "Turning Point 6," will simulate a massive earthquake leading to wide-scale destruction and large amounts of casualties.
Search and rescue units during drill (Archive photo: Roi Idan)
NATO Search and rescue units, as well as supporting units will arrive in Israel to take part in the drill, which will also include Israel's emergency services, the Police, Magen David Adom, fire services, as well as local authorities, governmental ministries, and representatives from the political echelon.
Two weeks ago, Minister for Home Front Defense Matan Vilnai met with NATO Deputy Secretary General Claudio Bisogniero in Brussels to discuss the details of the joint exercise.
During his visit, Vilnai also met with other senior members of the organization and officials from the European Union in order to discuss future cooperation between Israel and Europe on emergency and crisis management issues.
"We have a big interest in deepening and widening our ties with the Union and with NATO," said Vilnai, adding that "both Europe and Israel can benefit from steady cooperation on preparedness and response to emergencies and crises."
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text:
Home » State News » Naveen Patnaik met former Prime Minister Atal Bihari Vajpayee
Naveen Patnaik met former Prime Minister Atal Bihari Vajpayee
in State News April 24, 2017	0 999 Views
Bhubaneswar: 24/4/17;Chief Minister (CM) Naveen Patnaik today met former Prime Minister Atal Bihari Vajpayee 6 pm at his Krishna Menon Marg residence in New Delhi.
It was just a courtesy call on Vajpayee ji to inquire regarding his health. The visit has no political significance. Odisha can never forget Atal ji. It was simply a sudden plan to meet him. The visit was not in the itinerary, CM Patnaik told to media persons.Patnaik spent about 15 minutes with the former Prime Minister
Previous: BJP levelled serious charges against two senior BJD leaders for missing incident of the grandson of freedom fighter
Next: State Assembly will hold a special session on May 18 and 19
Hyderabad:21/1/22: Makers of SS Rajamouli's highly anticipated film RRR have released a statement in which they reveal that they are ready to release the film on March 18 if the situation ...

redpajama_set_name: RedPajamaCommonCrawl


Ever seen a situation where someone says that she is in a working relationship with a man, she brings him around, and everyone knows him. Later,in that same close period, you still see her with another man, having another relationship with him! That is double dating. Dating two or more men at the same time! What is your view on this matter? Is it an ideal thing to do?
Many people who engage in double dating most times do not make it known to their partners that they have someone else. If they feel that double dating is right, what stops them from letting the cat out of the bag? If they think that what they do is bad,why would they hide it from their partners? Every relationship that is intended to end in marriage must be built on trust! I now ask:'if you trust your relationship and partner is real,why would you be having another?
If you don't trust your partner, maturity demands that you call him and let out your fear. If you still cannot gain his trust, oooh let him go! Only then will you be free as he is,to engage in another. Double dating may make you think you are smart, but hurting and cheating on the one who loves you is an injurious abuse!!!


text:
Electrical Experimenter, August, 1919, page 372:
POCKET WIRELESS SOON, PREDICTS MARCONI OFFICIAL.
Godfrey Issacs says the English Marconi company, of which he is managing director, expects to have a commercial service of wireless telephones in operation in New York and London early next year.
The company hopes to make arrangements with New York and London telephone companies so that eventually the British and American wireless 'phone subscribers will be able to sit at their desks and "Hello, London" or "Hello, New York."
Isaacs foresees the day, not far distant, when pocket wireless telephones will be in wide use. A business man's secretary, walking along the street, Isaacs says, will hear a bell ring in his pocket, will put a receiver to his ear and hear "his master's voice" give him instructions, probably from an airplane hundreds of miles away.
The Daily Mail says that the British government probably will compel all commercial airplanes and airships to carry wireless telephone or telegraph equipment. Larger machines of both types may be compelled to carry both.
United States Early Radio History > Personal Communication by Wireless > Early Radio Development and Speculation


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KSV BWO is een sportvereniging uit Hengelo met afdelingen voor voetbal en handbal. De club telt ruim 1100 leden. Het eerste voetbalelftal speelt in de Tweede klasse zondag (seizoen 2020/21).

Historie
Op 1 juni 1929 werd vanuit de Onze Lieve Vrouwe Parochie in de Hengelose buurt "De Noork" de rooms-katholieke voetbalvereniging Blauw-Wit opgericht, dat met twee senioren- en één juniorenteam deelnam aan de competitie van de toenmalige R.K. Utrechtse Voetbalbond.

In 1940 werd deze bond opgeheven en er werd aansluiting gevonden bij de KNVB. Vanaf dat moment werd de naam veranderd in Blauw Wit Oost, kortweg BWO, omdat in Amsterdam reeds een bekende vereniging met de naam Blauw-Wit bestond.

Sportpark De Noork
De club heeft vijf wedstrijdvelden en twee trainingsvelden op sportpark De Noork, waarvan 2 kunstgrasvelden. Ook heeft BWO een nieuw clubgebouw en tevens nieuwe kleedkamers waar de vereniging sinds september 2012 de beschikking over heeft. Er ligt ook een geasfalteerd handbalveld, maar dit is buiten gebruik omdat de handbal naar een indoor locatie is verhuisd.

Competitieresultaten 1941–2018

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Bekende (oud-)spelers
 Frank Ensink
 Martijn Abbenhues
 Peet Bijen
 Daniël Nijhof
 Reuven Niemeijer

Handbal
Het eerste damesteam van BWO speelt in het seizoen 2020/2021 in de regionale eerste klasse. Sinds het seizoen 2020/2021 beschikt de vereniging over een herenteam doordat HHV Donar fuseerde met BWO. Het eerste herenteam van BWO nam hierdoor de plek in de hoofdklasse van Donar in.

Externe link
 Officiële website

BWO
Sport in Hengelo

redpajama_set_name: RedPajamaWikipedia


text:
Dive Brief
Cheesecake Factory misled on COVID-19 disclosures, SEC says
Published Dec. 5, 2020
Flickr; Port of San Diego
Dive Brief:
Restaurant chain The Cheesecake Factory falsely told investors it was operating sustainably, the Securities and Exchange Commission (SEC) said on Friday.
In its announcement, the agency said it had reached a settlement with the company on the charges.
"It is ... important that issuers who make materially false or misleading statements regarding the pandemic's impact on their business and operations be held accountable," SEC Chairman Jay Clayton said in a statement.
Dive Insight:
This is the SEC's first action charging a public company for misleading investors about the pandemic's financial impact.
The company in March and April publicly described itself as "operating sustainably" during the pandemic, a materially false and misleading statement, the SEC says, because internal documents showed it was losing $6 million in cash a week and had only 16 weeks of cash remaining.
Privately, the company shared its financial position with potential equity investors and lenders as it sought to boost its liquidity.
The company also didn't disclose it had told landlords it wouldn't pay rent in April. Instead, it said in its reporting it was taking steps to preserve its financial flexibility.
The disclosure failures violate reporting provisions of federal securities laws, the SEC says. Without admitting the findings in the order, the company agreed to pay a $125,000 penalty. The SEC says the penalty reflects the company's cooperation with the investigation.
"When public companies describe for investors the impact of COVID-19 on their business, they must speak accurately," said Stephanie Avakian, the SEC's director of enforcement.
SEC charges e-commerce startup with defrauding investors By Robert Freedman • Nov. 24, 2020
SEC penalizes Hilton $600K for disclosure lapse By Robert Freedman • Oct. 8, 2020
Using data-analytics tool, SEC catches two companies' accounting violations By Robert Freedman • Sept. 28, 2020
Filed Under: Financial reporting, Compliance
Latest in Financial reporting
Southwest ups tech spend to $1.3 billion
By Grace Noto


Testing out the WordPress Android app.
Don't think I'd want to do a huge blog post using this, but it works for short entries with cat pics.
T-Mobile about to unveil new phone based on Google's Android.
If you're at all curious what a cellphone based on Google's Android platform would be like then you may want to check out the live streaming T-Mobile G1 with Google Press Conference that's due to start in a few minutes at 10:30AM EST. This is the first phone based on Google's mobile OS and there's been lots of speculation on how big an impact it'll make in the market.
If you miss the live feed you'll be able to pull the video up on demand shortly afterwards at the same site. Just thought some of you would be interested in checking it out.


Would you like to spend less money on your Phone, Internet and TV? Well, you're in luck. If you get a Frontier Bundle in Warner Springs, CA, you will enjoy a discounted rate on all your services—and you only have to remember to pay one bill. Whether you just want DISH TV and Internet, or Internet and Home Phone, we can help you get what you need. Give us a call now to find the best Frontier deals in The Golden State.
It's a great time to switch from your old cable provider in Warner Springs to something new. Frontier Internet is available for installation in your Warner Springs, CA home. Frontier Internet can give you awesome speeds as fast as 1 Gig (in limited areas), upload and 1 Gig (in limited areas), download, which is more than enough bandwidth for almost every online activity. Frontier Internet will even lock in your price for three years without making you sign a contract. When you sign up for a Frontier Internet package, you'll even get a Wireless Router so you can create an in-home Wi-Fi network. Don't wait to call and find out what Frontier Internet deals are available in your area.


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ENVIRONMENT Environment
HMS Investigator Discovery: A Rewarding Find
30 July 2010 CBC News	0 Comments
Parks Canada archeologists who found HMS Investigator, a British naval ship that was stranded in the Northwest Passage 155 years ago, are equally pleased to have discovered the graves of three of the ship's crew.
The wreck of Investigator was discovered this week at the bottom of Mercy Bay, just off Banks Island in the Northwest Territories. The site of the wreck is in Aulavik National Park.
Parks Canada archeologists located the wreck shortly after they started a sonar scan of Mercy Bay on Sunday, according to officials.
On Monday, another Parks Canada team that was searching on land found the gravesites of three Investigator crew members who died of scurvy in April 1853.
"Just the number of finds within just a short period of time has been really rewarding, from the discovery of the ship to the discovery of the graves," Henry Cary, a Parks Canada archeologist who took part in the land search, told CBC News on Thursday.
Trapped in ice
The wreck of HMS Investigator, discovered at the bottom of Mercy Bay this week, stands upright in about 11 metres of water, with the upper deck under about eight metres of water. (Parks Canada)Investigator, captained by Robert McClure, was dispatched in 1850 to search for Sir John Franklin's crew and their two ships, HMS Erebus and HMS Terror, after they disappeared in the Northwest Passage.
The crew were trapped in the ice at Mercy Bay for more than two years, during which time three members — two able seamen and the gunner's mate — died. The 63 who survived were eventually rescued by a Royal Navy sledge team that took them to another ship.
Before leaving their ship behind, the Investigator crew buried much of the ship's provisions on Banks Island. Cary's group was conducting the land search to explore the ship's cache site, as well as to locate the crew members' graves.
Cary said historians have known about the deceased men and when they died, but no one really knew where they were buried.
A clue that led the archeologists to the gravesites was found in the recent translation of a diary from a German missionary who had accompanied McClure's expedition.
An original translation of the diary said the men were buried on the shore, but Cary said his team followed the more recent translation, which suggested the men were buried farther inland.
May be in good condition
Officials estimate that the graves, which are in undisturbed condition, are about 60 metres from the Investigator cache site.
"They're on the plains just coming up from the beach towards the cache site, and they just look exactly like you'd imagine a freshly dug grave shaft to be, with evidence of the cuts around the sort of oval-shaped boundaries, and then the earth mounded in the centre," Cary said.
The fact that there are mounds of earth on the graves suggest that the bodies underneath may still be in good condition, Cary added.
"Generally, what happens is as the bodies decompose, the earth begins to collapse in," he said. "And so instead of getting a mound, you get a divot or a slumping."
Edward Eastaugh, an anthropology lab manager from the University of Western Ontario, used a magnetometer — a sophisticated metal detector of sorts — that showed the presence of some kind of metal in the graves.
That metal could be from nails or tacks on the men's coffins, or they could be copper name plates affixed to the coffins, Cary said.
No exhumation plans
Parks Canada has contacted the British government about the discovery of HMS Investigator and the three gravesites, but officials say the sites will remain undisturbed for now.
"Any excavation of these remains would be in the future — certainly not this season, and maybe not next, depending on how we go about deciding whether we are going to excavate them," Cary said.
"Any future work that we would do on the burials would be approached with extreme respect and caution. We would ensure that there would be a series of consultations and permitting and just going through the whole process to ensure that those remains are treated as respectfully as possible."
Cary, Eastaugh and the rest of the land crew have until Aug. 3 to finish surveying the site of the graves and the cache for other hidden artifacts.
"The remains themselves are quite extensive, but we're probably the first to do a real systematic survey of artifacts on the surface," Cary said.
"Previous groups have not had a huge amount of time, just because of the remoteness of the site, to go over the site over and over again looking for surface finds and different features. So there's several new discoveries that way."
Seeking Erebus and Terror
The goal of Robert McClure and those aboard HMS Investigator in 1850 was to find out what happened with Sir John Franklin's ill-fated expedition to the Northwest Passage.
This week's discovery comes as Parks Canada resumes a three-year federally funded mission to locate Franklin's ships, HMS Erebus and HMS Terror. Searchers will head to Queen Maud Gulf in August.
Marc-André Bernier, the head of Parks Canada's underwater archeology service, said this week's discovery of HMS Investigator raises their hopes of finally finding Franklin's long-lost ships.
"It raises the optimism because those are very difficult projects, and to find something like this actually shows that it is feasible," Bernier told reporters on Wednesday.
"You know, the search for the Erebus and Terror has been going on for 160 years, and a lot of people have taken a shot at it. … We see it as this is huge puzzle for which every single piece counts."
In 2008, the Parks Canada team found small remnants of copper sheets that may have been part of Erebus and Terror. Search efforts were put on hold in 2009 because a Canadian Coast Guard icebreaker was not available.
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For more news from Canada visit CBC News.


text:
Home Things Scourers
As time went on, the Scourers became increasingly corrupt. Far away from the jurisdiction of their native magical governments, many indulged a love of authority and cruelty unjustified by their mission. Such Scourers enjoyed bloodshed and torture, and even went so far as trafficking their fellow wizards.
-- from the History of Magic in North America (Pm)
17th century – Rogue band of vengeful mercenary wizards who caused a huge problem between wizards and No-Majs in North America.
During the early years of the European Wizarding community in America, there was no government or laws. As a result, unscrupulous bands of wizards formed mercenary bands to enforce their brand of justice. They were bounty hunters who would search for not only outlaws and criminals but also any witch or wizard for whom someone was willing to pay gold. Over the years in the late 1600s, the Scourers became feared for the brutal, dangerous groups that they were. They were not above capturing and accusing innocent No-Majs as magical in order to collect bounties during the years of witch hysteria.
During the Salem Witch Trials of 1692-93, two of the Puritan judges were actually Scourers settling the score in feuds with fellow Wizards.
Due to the trauma of the Salem Witch Trials, the Magical Congress of the United States (MACUSA) was formed in 1693. This secret government separate from the No-Maj's was first led by President Josiah Jackson, who hired a new group of Aurors to track and hunt down Scourers as well as the criminal fugitives from European wizardry who helped Scourers identify and persecute witches and wizards (Pm:MACUSA).
Josiah Jackson
Charity Wilkinson
Theodard Fontaine
Agilbert Fontaine
Gondulphus Graves
Abraham Potter
Isolt Sayre
MACUSA headquarters in the Appalachian Mountains
Woolworth Building
Flame-Freezing Charm
MACUSA founded
MACUSA relocates to Williamsburg
MACUSA relocates to Baltimore
MACUSA relocates to Washington
MACUSA relocates to New York
American Witches and Wizards Engage in "Country or Kind" Debate
Great Sasquatch Rebellion of 1892
Department of Magical Law Enforcement (MACUSA)
Magical Congress of the United States of America (MACUSA)
Aurors
What is the US equivalent of the Ministry of Magic?
"scour" - scrub clean
Scourer is a real dictionary word with two meanings - one, someone who scours or cleans or purges [http://www.merriam-webster.com/dictionary/scourer] (and their descendants were into ethnic cleansing in that they wanted to exterminate wizards). But there is another obscure meaning traveler [https://www.vocabulary.com/dictionary/scourer] rover, footpad, prowling robber [https://www.wordnik.com/words/scourer][https://en.wiktionary.org/wiki/scourer].
Since JKR used the word "mercenaries" the scourers would have started off as soldiers of fortune - fighting for whomever paid them. Then they evolved into bounty hunters - probably because the skillset was transferable. And a couple of them even became judges - those who got to decide who was guilty of witchcraft and who wasn't - what influenced said decisions is only hinted at. We know that they were generally motivated by money, but all we know for sure is that they were known to use their position to settle old scores.
Two reasons are given for scourers teaching their descendants that magic is real (ie recognizing the signs) and evil - vengeance and avoiding discovery. If the scourers were discovered, they would be charged and sentenced - so they did have something legitimately to fear from the presence of magic. -vaml
Pottermore: Magical Congress of the United States (MACUSA) by J.K. Rowling
Pottermore: History of Magic in North America by J.K. Rowling
Wikipedia: Salem Witch Trials
Smithsonian Institution: A Brief History of the Salem Witch Trials
Salem Witch Museum
Tags: revenge search/searching
Editor: Steve VanderArk
Acolytes Advance Guard Argentinian Council of Magic Brotherhood of Goblins (B.O.G.) Campaign for Greater Freedom for Wizards Charms Club Dark Force Defence League Death Eaters Department of Magical Law Enforcement (MACUSA) Duelling Club Dumbledore's Army (The D.A.) Fresh Air Refreshes Totally (F.A.R.T.) Ghosts Council at Hogwarts Goblin Rights groups Gobstones Club Hogwarts Board of Governors Inquisitorial Squad International Association of Quidditch
International Confederation of Wizards (I.C.W.) International Confederation of Wizards Quidditch Committee (I.C.W.Q.C.) International Symposium of Animagi Investigative Team Magical Congress of the United States of America (MACUSA) Major Investigation Department Medieval Assembly of European Wizards Ministry of Magic Ministère Des Affaires Magiques Most Extraordinary Society of Potioneers Mudblood Death Camps National Gobstone Association Opposition Order of Merlin Order of the Phoenix Pocomtuc tribe Pure-blood Extremist Groups Quidditch Union for the Administration and Betterment of the British League and its Endeavors (Q.U.A.B.B.L.E.)
Rotfang Conspiracy Salem Witches' Institute Scourers Second Salem a.k.a. New Salem Philanthropic Organization Second Salemers Silver Spears Slug Club Snallygaster Protection League Society for Distressed Witches Society for the Promotion of Elfish Welfare (S.P.E.W.) Society for the Reformation of Hags Society for the Support of Squibs (S.S.S.) Society for the Tolerance of Vampires (S.T.V.) Troll Rights Movement Wizarding Supporters of Scottish Rugby Union (W.S.S.R.U.) Wizards' Council
1692 : Salem Witch Trials
1693 : MACUSA founded
1760 : MACUSA relocates to Williamsburg
the 1770s : MACUSA relocates to Baltimore
circa 1776 : MACUSA relocates to Washington
1777 : American Witches and Wizards Engage in "Country or Kind" Debate
1892 : Great Sasquatch Rebellion of 1892
circa 1900 : MACUSA relocates to New York


Raking in a whopping Rs 55.8 million at the Box Office just five days into its release, Loot 2, the sequel to the 2012 blockbuster, seems on course to shatter records, despite receiving mixed reviews from audiences.
According to Madhav Wagle, one of the producers, the film collected a total of Rs 1.51 crores on the first day of its release, on Friday, which made Loot 2 the highest grossing Nepali movie on its opening day. The film went on to collect Rs 1.61 crores on Saturday, Rs 85 lakhs on Sunday, and Rs 96 lakhs on Monday. The collection on Tuesday dipped to Rs 65 lakhs.
Loot 2 features actors Saugat Malla, Dayahang Rai, Karma, Bipin Karki, Srijana Subba and Alisha Rai, among others.


Boston, MA: In a new book released this month, United for a Fair Economy and Responsible Wealth urge a more honest national dialogue about what makes wealth and success possible, and in doing so, a healthier debate about the public policy choices before us.
The Self-Made Myth—And the Truth About How Government Helps Individuals and Businesses Succeed, by United for a Fair Economy executive director Brian Miller and Responsible Wealth project director Mike Lapham, exposes the false claim that business success is solely the result of the heroic effort of a single individual. The authors contend that, among other supports, businesses are built atop public structures and services established and maintained by taxpayers' collective investments made through government. As such, those who have benefitted the most from those public supports owe something back to the society that made them possible.
The Self-Made Myth book tour launched last week, fittingly at a public institution of learning, the Boston Public Library. Miller and Lapham will be joined in select cities by individuals profiled in the book.
United for a Fair Economy (UFE) is a national organization working to close the growing income and wealth divides in the U.S. Responsible Wealth, a project of UFE, is a network of over 700 business leaders, high-wealth and upper-income advocates of progressive tax policies and corporate accountability. Learn more about The Self-Made Myth at www.selfmademyth.org.


text:
\section{Introduction}
\label{sec:Introduction}
\par
\medskip

Self-organizing spatial pattern formation in interacting population
models is an important area of research to understand the
distribution of the species within their habitats. Mainly
reaction-diffusion equation models are studied analytically with
supportive numerical simulations to understand the nature of
population patches and whether they are changing with respect to
time or not. Investigations for various types of Turing and
non-Turing pattern formations in two dimensional prey-predator
models have received significant attention \cite{alonso2002mutual, banerjee2011self,
baurmann2007instabilities, medvinsky2002spatiotemporal, Wang2007spatio}. Spatio-temporal prey-predator models exhibit various types of self-organizing spatial patterns; either
they are stationary or non-stationary. Stationary patterns are
characterized by heterogeneous distribution of populations over
space which do not change with time once they reach the non-constant
steady-state. On the other hand, the non-stationary patterns
change with time continuously and never reach any non-constant
steady-state. The stationary patterns produced by the
spatio-temporal prey-predator models for two dimensional spatial
domains can be classified as: (i) spot patterns
(cold-spots and hot-spots), (ii) labyrinthine/stripe patterns, or (iii) the mixture of
spot and stripe patterns. These three types of patterns are mostly
observed for parameter values taken from Turing-domain or
Turing-Hopf-domain, when  the  diffusivity of two species differ by a
significant magnitude. Sometimes one does find labyrinthine patterns
as a non-stationary pattern but for parameter values well inside
the Turing-Hopf domain. The non-stationary patterns may include periodic
patterns (sometimes as periodic traveling waves), interacting
spiral patterns and chaotic patterns. The occurrences of
spatio-temporal chaos for parameter values lying within the Turing
domain or outside the Turing-domain remains a debatable and controversial issue
\cite{malchow2007spatiotemporal}. There is no unified approach in determining the
parameter values in a systematic way or in identifying the mechanism which
leads to the spatio-temporal chaotic patterns \cite{banerjee2011self,
banerjee2012turing}. Derivation of necessary-sufficient condition(s),
either in terms of parametric restrictions or system
characterization for the onset of spatio-temporal chaotic patterns
for prey-predator type interacting models  thus remains an open
problem.

Recently, the role of environmental noise on spatio-temporal pattern
formation has been studied by the researchers for the noise added
spatio-temporal models of interacting populations
\cite{malchow2004oscillations, sun2009role, Wang2011color, li2012pattern}. In most of the cases, the
authors have been used temporally correlated and spatially
uncorrelated coloured noise terms. Although in some investigations
the alteration of spatial patterns due to the presence of additive
coloured noise terms have been reported but this type of formulation
is not ecologically viable as they correspond to spontaneous
generation or inhibition of populations at the places with no
population or very low concentration of population. On contrary, the
consideration of multiplicative noise terms signify the noisy
variation in some system parameters involved with the reaction
kinetics. It is evident that due to the environmental fluctuations,
the carrying capacity, birth and death rates, intensity of intra-
and inter-species competition rates, predation rates exhibit random
fluctuations to a modular extent around some average value
\cite{may1972limit}. Perturbation of one or more rate constants by the noise
terms leads to spatio-temporal models with multiplicative noise
terms. Several noise induced phenomena like noise-induced phase
transitions, noise enhanced stability/instability, noise delayed
extinction, stochastic resonance are investigated for the
spatio-temporal ecological models considered within fluctuating
environments. These analytical/numerical findings provide important
information towards how the species establish within their respective habitats
or help us to identify the important risk factors leading towards the
extinction of species.

The main objective of this paper is to study a simple model of
predator-prey with ratio-dependent functional response and density
dependent death rate of predator, along with the self-diffusion terms corresponding to the random movement of the individuals within two dimensional habitats, and study the influence of small amplitude
heterogeneous perturbations to the linear intrinsic growth rates.

\section{The spatiotemporal model}
\label{sec:Basic Model}

\medskip

\setcounter{equation}{0}

Consider that $u \equiv u(x,y,t)$ and $v \equiv v(x,y,t)$ represent the prey and predator population densities at any time $ t $ and at  the spatial location $ (x,y)\in \Omega $, where $ \Omega \in R^{2}$ is a square domain in $\mathbb{R}$ with boundary $\partial{ \Omega} $.
The spatiotemporal dynamics of prey-predator interaction with logistic growth for prey, ratio-dependent functional response and density dependent death rate for predators is governed by the following system of nonlinear coupled partial differential equations (PDE), in terms of dimensionless variables and parameters:

\begin{eqnarray}
\frac{\partial u}{\partial t} & = & u(1-u) -\frac{\alpha uv}{u+v} + \nabla^{2} u \equiv f(u,v)+\nabla^{2}u,         \label{eq1}
\\
\frac{\partial v}{\partial t} & = & \frac{\beta uv}{u+v} - \gamma v - \delta v^{2} +d\nabla^{2} v \equiv g(u,v)+\nabla^{2}v
\label{eq2}
\end{eqnarray}
subjected to positive initial conditions:
\begin{eqnarray}
 u(x,y,0)\equiv u_0(x,y)>0, v(x,y,0)\equiv v_0(x,y)>0,\text{for all} (x,y)\in\Omega
\label{eq:equation3}
\end{eqnarray}
and zero-flux boundary conditions:
\begin{eqnarray}
 \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0, \text{ for all} (x,y)\in \partial{ \Omega}, t>0
\label{eq:equation4}
\end{eqnarray}
where $\nu$ is the outward drawn unit normal vector on the boundary, and $ \alpha$, $\beta$, $\gamma$ and $\delta$ are positive dimensionless parameters. The ratio of diffusivity of two species is denoted as $ d $.
The diffusion terms in this reaction-diffusion system for the prey and predator population are introduced to take care of the random movements of the individuals within two dimensional habitats. The zero-flux boundary condition indicates that predator-prey system is self-contained within the two dimensional habitat with no population flux across the boundary.
\subsection{Turing Bifurcation}
\par
\medskip

The ecologically feasible and coexisting equlibrium point for the correspoding temporal model are spatially homogeneous steady state for the reaction diffusion model (\ref{eq1}-\ref{eq2}) and can be obtained by solving the following two algebric equations, with $u,v>0$:
\begin{eqnarray}
    f(u,v) & = & u(1-u) -\frac{\alpha uv}{u+v}  = 0, \label{eq3} \\
    g(u,v) & = & \frac{\beta uv}{u+v} - \gamma v - \delta v^{2}  =  0.
\label{eq4}
\end{eqnarray}
If we denote the coexisting equilibrium point by $E_*\left(u_{*},v_{*}\right)$, then $u_*$ and $v_*$ satisfy the equtions $\displaystyle 1-u -\frac{\alpha v}{u+v}\,=\,0\,=\,\frac{\beta u}{u+v}-\gamma -\delta v$. The explicit expression for the components of $E_*$, conditions for their feasible existence and possible number of coexisting steady-states are discussed in Ref.~\cite{banerjee2012turing} extensively. Further we assume that $E_*$ is coexisting  homogeneous steady-state which is locally asymptotically stable for the temporal counterpart of the system
(\ref{eq1}-\ref{eq2}), i.e., the conditions $ a_{11} + a_{22}  <  0 $ and $ a_{11} a_{22} - a_{12} a_{21} >  0 $ are simultaneously satisfied, where $a_{11}, a_{12}, a_{21} $ and $ a_{22}$ are given by
\begin{eqnarray*}
    a_{11} \,=\,  1 -2u_* -\frac{\alpha v_*^{2}}{(u_*+v_*)^{2}}, \,\,\,
    a_{12} \,=\,  - \frac{\alpha u_*^{2}}{(u_*+v_*)^{2}}, \,\,\,
    a_{21}  \,=\, \frac{\beta v_*^{2}}{(u_*+v_*)^{2}}, \\
    a_{22} \,=\,  -\gamma-2\delta v_* + \frac{\beta u_*^{2}}{(u_*+v_*)^{2}}. \hspace{1.5in}
\end{eqnarray*}
\par

The basic criteria for the diffusion-driven instability or Turing instability is that the stable homogeneous steady-state becomes unstable due to small amplitude heterogeneous perturbation around the homogeneous steady-state. Hence for the occurrence of Turing instability, first we need a locally stable coexisting steady-state for the corresponding temporal model. The linear stability analysis for spatiotemporal model (\ref{eq1}-\ref{eq2}) around the homogeneous steady-state $E_*$ and the conditions required for the onset of Turing-instability are well established and used by several researchers. Here, we just mention the three basic mathematical criteria required for Turing-instability of the model (\ref{eq1}-\ref{eq2}) around $E_* (u_{*}, v_{*} )$, as follows:\\
\begin{eqnarray}
    a_{11} + a_{22} & < & 0, \label{Tur1}\\
    a_{11} a_{22} - a_{12} a_{21}  & > & 0,  \label{Tur2}        \\
    da_{11} + a_{22}  & > &2\sqrt{d}\sqrt{a_{11} a_{22}- a_{12} a_{21} }, \label{Tur3}
\end{eqnarray}
and Turing-instability sets in at the critical wave number
\begin{eqnarray}
    k_{cr}^2 = \frac{d a_{11} + a_{22}}{2\sqrt{d} } > 0.
\label{Turkcritical}
\end{eqnarray}
Satisfaction of the condition (\ref{Tur3}) implies homogeneous steady-state becomes unstable under small amplitude heterogeneous perturbation for wavenumbers $k>k_{cr}$ and Turing pattern emerges. Type(s) of emerging spatial pattern(s) (as mentioned earlier in the introduction), depend upon the specific choice of parametric values and magnitude of $ d $.
\par The Turing-bifurcation curve is defined by the following equation
$$ da_{11} + a_{22}  \,=\, 2\sqrt{d}\sqrt{a_{11} a_{22}- a_{12} a_{21} }.$$
\par Solving this equation for $  d $, we can find the critical threshold for the ratio of diffusivity $d_{cr}$, above which the Turing patterns emerge. In the next section we are going to report the varieties of spatial patterns generated by the system (\ref{eq1}-\ref{eq2}) for suitable choice of
the parameter values satisfying the Turing bifurcation conditions.
\section{Self-organizing spatial patterns}
\medskip

In this section, we present the numerical simulation results and varieties of spatial patterns  that are exhibited by the spatiotemporal model (\ref{eq1}-\ref{eq2}) for parameter values
within the Turing and Turing-Hopf domain. If we take the parameter
values $\alpha = 2.0$, $\beta = 1$, $\gamma = 0.6$ and $\delta =
0.1$ then we find the unique coexisting equilibrium point
$E_*(0.2287,0.1436)$ which is a homogeneous steady-state for the
system (\ref{eq1}-\ref{eq2}). $E_*$ is locally asymptotically stable
for the temporal counterpart as the conditions (\ref{Tur1}) and
(\ref{Tur2}) are satisfied. In order to understand the role of
diffusivity of the prey and the predator towards the destabilization
of the homogeneous steady-state we need to find the quantity
$d_{cr}$ as mentioned in the previous section. For the chosen set of parameter
values, we find $d_{cr}=4.9157$ and hence the heterogeneous
perturbations lead to Turing patterns for $d>d_{cr}$.

\par The existence and non-existence of Turing patterns are solely
dependent upon the magnitude of the parameters involved with the
model under consideration. Here we consider $\alpha$ and $d$ as the
bifurcation parameters to construct the Turing bifurcation diagram.
We have presented the Turing bifurcation diagram in $\alpha
d$-parametric plane (see~Fig.~\ref{fig:1}). The bifurcation
diagram is prepared keeping $\beta = 1$, $\gamma = 0.6 $ and $\delta
= 0.1$ fixed. $\alpha$ and $d$ are considered as the controlling
parameters to obtain different spatiotemporal patterns. We have
presented three bifurcation curves in the bifurcation diagram,
namely Turing-bifurcation curve (blue curve), temporal
Hopf-bifurcation curve (red dashed line) and temporal homoclinic
bifurcation curve (black dotted curve). The coexisting equilibrium point
$E_{*}$ for the temporal model, corresponding to the system
(\ref{eq1}-\ref{eq2}), is stable for $\alpha<\alpha_{h}$ and it
looses stability through the Hopf-bifurcation at $\alpha_{h} = 2.01$
(approx). The Turing instability region is the region lying above
the Turing bifurcation curve and which is divided into two parts by
the vertical line $\alpha=\alpha_{h}$. Part of the Turing domain
lying in the region $\alpha>\alpha_{h}$ is the Turing-Hopf domain
where temporal and spatiotemporal perturbations are both unstable.

\par For the chosen set of parameter values we find only one feasible
coexisting equilibrium point $E_{*}$ therefore we will discuss only
the spatial-pattern formation around this homogeneous steady-state.
We have obtained various patterns generated by the model under
consideration through exhaustive numerical simulations. All
numerical simulations are carried out over a $200\times200$ lattice
with time-step $\Delta t = 0.01$ and spatial steps $\Delta x =
\Delta y = 2$. All numerical simulations are performed using the
forward Euler method for the temporal part and five point finite
difference scheme for the diffusion part. A small amplitude
heterogeneous perturbations to the homogeneous steady-state are
introduced by $ u_{0}\left( x_{i},y_{j}\right) = u_* + \epsilon
\xi_{ij}$, $ v_{0}\left( x_{i},y_{j}\right) = v_* + \epsilon
\eta_{ij}$ where $ \epsilon = 0.001 $, $\xi_{ij}$ and  $\eta_{ij}$
$(i, j = 1, 2)$ are spatially uncorrelated Gaussian white noise
terms. We have observed four different types of patterns within the
Turing domain, cold-spot, labyrinthine, mixture of spot and stripe
and spatiotemporal chaotic patterns.

\begin{figure}%
\leavevmode \centering
\includegraphics[]{Tboundary.pdf}
\caption{\footnotesize Five different types of spatial patterns
observed for the parameter values lying within the Turing
instability region (above the blue curve which is Turing bifurcation
curve) and below Turing instability region. Choices of parameter values are marked with four different symbols based upon the resulting patterns:
\textcolor{red}{${\circ}$} cold-spot, \textcolor{green}{$\Box $}
mixture of spot-stripe, \textcolor{blue}{$\diamond$} labyrinthine, \textcolor{magenta}{$\ast$} chaotic and \textcolor{red}{${+}$} interacting spiral. Hopf-bifurcation curve (shown by red dashed line) and temporal homoclinic
bifurcation curve (shown by black dotted curve).}\label{fig:1}
\end{figure}

\par We have observed only cold-spot and mixture of spot-stripe patterns
for parameter values within the Turing domain, whereas mixture of
spot-stripe, labyrinthine and chaotic patterns are observed for
parameters lying in the Turing-Hopf domain. Numerical simulations have been  performed for systematic choices of $\alpha$ and $d$ within the Turing domain in order to understand the entire variety of spatiotemporal patterns for parameter values within the Turing domain. Obtained patterns for different combinations of $\alpha$ and
$d$ are marked with four different symbols. The whole regime of
patterns in the Turing domain is shown in the Fig.~\ref{fig:1}.

\begin{figure}%
\leavevmode \centering
\mbox{\subfigure[]{\includegraphics[width=2.5in,height=2in]{alpha_195_d_10}}
\subfigure[]{\includegraphics[width=2.5in,height=2in]{alpha_205_d_10}}}
\mbox{\subfigure[]{\includegraphics[width=2.5in,height=2in]{alpha_215_d_10}}
\subfigure[]{\includegraphics[width=2.5in,height=2in]{alpha_230_d_10}}}
\mbox{\subfigure[]{\includegraphics[width=2.5in,height=2in]{alpha_215_d_1}}}
\caption{\footnotesize Spatial distribution of prey ($ u $) obtained at
$t=1000$ for $\beta = 1$, $\gamma = 0.6$, $\delta = 0.1$, $d =10.0$
and (a) $ \alpha = 1.95$; (b) $ \alpha = 2.05 $; (c) $ \alpha = 2.15
$; (d) $ \alpha = 2.30$; and (e) $ d = 1.0$, $\alpha = 2.15$. The color bar in each panel represents the magnitude of $ u $.} \label{fig:2}
\end{figure}

\par A sample of each type of patterns observed within the Turing domain
are shown in Fig.~\ref{fig:2}. At the lower panel we have presented
the interacting spiral pattern which is observed for parameter
values just outside the Turing-Hopf domain and for $d=1.0$. As the
Turing instability can not occur for $d=1.0$, we can label this
pattern as non-Turing pattern. To understand the stationary nature
of the observed patterns, we have calculated the time evolution of
spatial average for the prey and predator densities against time
corresponding to the patterns reported in Fig.~\ref{fig:2} and the
results are presented in Fig.~\ref{fig:3}. The time evolution of
$\left\langle u \right\rangle $ and $\left\langle v \right\rangle $ are considered over a longer period of time to
ensure that the final observations are far away from the transient
behavior. Looking at the time evolution of spatial averages
presented at Fig.~\ref{fig:3}, one can easily understand that
cold-spot, mixture of spot-stripe and labyrinthine patterns are
stationary patterns whereas the interacting spiral pattern is a
non-stationary pattern. The time evolution of the spatial averages
are continuously oscillating and never settle down to any stationary
value. Corresponding to the spatiotemporal chaotic pattern presented
in Fig.~\ref{fig:2}(d), the time evolution of spatial averages
exhibit irregular oscillations for $t>650$ and the irregularity
sustains at all future time. Here we have presented the results up
to $t=1000$ from the sake of brevity.

\begin{figure}%
\leavevmode \centering
\mbox{\subfigure[]{\includegraphics[width=2.5in,height=2in]{alpha_195_d_10_avg}}
\subfigure[]{\includegraphics[width=2.5in,height=2in]{alpha_205_d_10_avg}}}
\mbox{\subfigure[]{\includegraphics[width=2.5in,height=2in]{alpha_215_d_10_avg}}
\subfigure[]{\includegraphics[width=2.5in,height=2in]{alpha_230_d_10_avg}}}
\mbox{\subfigure[]{\includegraphics[width=2.5in,height=2in]{alpha_215_d_1_avg}}}
\caption{\footnotesize Time evolution of spatial averages of the
prey $\left\langle u \right\rangle $ and predator $\left\langle v \right\rangle $ populations, presented in (dark) blue and and (light) green colors,
respectively, for $t\in[0,1000]$ corresponding to the patterns
presented in Fig.~\ref{fig:2}.}\label{fig:3}
\end{figure}

\section{Noise Added Model}
\par In this section we are interested to study the effect of noise
on the spatiotemporal pattern formation for the model
(\ref{eq1}-\ref{eq2}). For this purpose we introduce uncorrelated
multiplicative white noise terms to the growth equations of both the
prey and the predator populations, accordingly the noise added model
is given by:
\begin{eqnarray}\label{noisemodel1}
\frac{\partial u}{\partial t}=& u\left(1-u\right) - \frac{\alpha uv}{u+v} + \nabla^2 u + \sigma_{1}u\xi_{1}(t,x,y),\\
\label{noisemodel2} \frac{\partial v}{\partial t}=& \frac{\beta
uv}{u+v} - \gamma v -\delta v^2 + \nabla^2 v +
\sigma_{2}u\xi_{2}(t,x,y),
\end{eqnarray}
where $\xi_{1}(t,x,y)$ and $\xi_{2}(t,x,y)$ are two temporally as
well as spatially uncorrelated Gaussian white noise terms. The
averages and correlations of the noise functions are given by,
$$ \left\langle \xi_{1}\left(t,x,y\right)\right\rangle = \left\langle\xi_{2}\left(t,x,y\right)\right\rangle = 0$$
$$\left\langle \xi_{j}\left(t,x,y\right)\xi_{j}\left(t_{1},x_{1},y_{1}\right)\right\rangle
= 2\sigma_{j}\delta\left(t - t_{1}\right)\delta\left(x -
x_{1}\right)\delta\left(y - y_{1}\right), j = 1,2,$$ where $
\sigma_{1}$ and $\sigma_{2}$ are the intensities of the
environmental driving forces. The introduction of the noise terms
can be justified from ecological point of view also. Here we are
considering the dimensionless model and hence we can see from
(\ref{eq1}-\ref{eq2}) that the intrinsic growth rate of the prey is
`1' and the intrinsic death rate for the predators is `$\gamma$'.
Now we assume that the intrinsic growth rate of the prey and the
death rate of the predator population are subjected to environmental
fluctuations and hence we can perturb them by spatiotemporal noise
terms. As a result, if we consider the introduction of noise terms
into the said rate constants, that is, $1\rightarrow
1+\sigma_1\xi_1(t,x,y)$ and $\gamma\rightarrow
\gamma+\sigma_2\xi_2(t,x,y)$ then we find the noise added model
(\ref{noisemodel1}) - (\ref{noisemodel2}) from the original
spatiotemporal model (\ref{eq1}-\ref{eq2}).

\begin{figure}
\leavevmode \centering
\mbox{\subfigure[]{\includegraphics[width=7.0cm]{w_noise_pat}}
\subfigure[]{\includegraphics[width=7.0cm]{noise_0010_pat}}}
\mbox{\subfigure[]{\includegraphics[width=7.0cm]{w_noise_avg}}
\subfigure[]{\includegraphics[width=7.0cm]{noise_0010_avg}}}
\caption{\footnotesize Spatial distribution of prey obtained at
$t=3000$ for $\alpha = 2.15, \beta = 1,\gamma = 0.6, \delta = 0.1$
and $ d =8.5$ (a) $\sigma = 0$ (b) $\sigma = 0.01$; the color bar in each panel represents the magnitude of $ u $. Time evolution
of spatial averages of prey $\left\langle u \right\rangle $ and predator $\left\langle v \right\rangle $ populations   in (dark) blue and and (light) green colors,
respectively, for (c) $\sigma = 0$, and (d)
$\sigma = 0.01$.} \label{fig:4}
\end{figure}

\par In order to understand the effect of noise on the spatiotemporal
pattern formation, we have performed numerical simulations with the
changing magnitude of noise intensities. Here we consider the
parameter set $\alpha = 2.15$, $\beta = 1$, $\gamma = 0.6$, $\delta
= 0.1$, $ d = 8.5$ and assume that both the noise intensities are
same, that is $\sigma_{1}= \sigma_{2} = \sigma$. The numerical
simulation is performed for $\sigma=0.01$ and the resulting
distribution of the prey population is shown at Fig.~\ref{fig:4}(b)
along with the pattern observed for $\sigma=0$ (see
Fig.~\ref{fig:4}(a)). There is no significant difference in the
reported patterns as the noise intensity is small. We have also
calculated the spatial averages for the prey and predator
populations up to $t=3000$ for $\sigma=0$ and $\sigma=0.0010$ and
the results are presented at Fig.~\ref{fig:4}(c)-(d). It is clear
from these figures that the small noise does not change the system
characteristics when the spatiotemporal patterns are oscillatory in
nature. One can observe similar results for the parameter values
corresponding to the stationary patterns and when noise intensities
are not very high.

\par In order to support our claim, we have prepared a bifurcation
diagram for the same set of parameter values, as mentioned above,
and considering the ratio of diffusivity as the bifurcation
parameter. The bifurcation diagrams are prepared for the
spatiotemporal model (\ref{eq1}-\ref{eq2}) as well as for the noise
added model (\ref{noisemodel1}) - (\ref{noisemodel2}) with
$\sigma=0.01$, see Fig.~\ref{fig:5}. We have performed the
numerical simulations for $d=1.0$ to $d=11.0$ with increments of $0.5$ and
spatial averages are calculated for the prey population up to
$t=3000$. After discarding the initial transients, all local maxima
and minima of the spatial average of the prey population are plotted
against each values of `$d$'. If we compare the bifurcation
diagrams, then it is evident that the number of local maxima and
minima increases with the introduction of small amplitude noise
terms mainly within the parameter regime where we find
non-stationary spatial patterns in the absence of noise. One natural
question arises whether the bifurcation diagram presented at the
right panel of Fig.~\ref{fig:5} changes from one simulation to
other. It does not change as we have taken the spatial average at
each time step and the spatial domain is sufficiently large to avoid
such kind of variation in the bifurcation diagram.

\begin{figure}
\leavevmode \centering
\mbox{\subfigure[]{\includegraphics[width=7.0cm]{bifur_WON_alpha_215}}
\subfigure[]{\includegraphics[width=7.0cm]{bifur_Noise_001_alpha_215}}}
\caption{\footnotesize Bifurcation diagram with respect to
bifurcation parameter $d$ with other parameters remaining same as in the
previous figure for (a) $\sigma = 0$ and (b) $\sigma = 0.01$.}
\label{fig:5}
\end{figure}

\par After understanding the effect of small intensity noise on the
resulting patterns, it is important to understand what happens with
the increasing magnitude of noise intensities. Of course, we can
find some noisy patterns when the magnitude of $\sigma$ is
significantly large and some times it is difficult to identify the
particular type of patterns they are representing. In order to avoid
this type of confusion and appropriate understanding of the
underlying phenomena, we have calculated the signal-to-noise-ratio
(SNR) against a range of noise intensities. Keeping $ \alpha =
2.15$, $\beta = 1$, $\gamma = 0.6$, $\delta = 0.1$, $d = 8.5$ fixed,
we have performed the numerical simulations for a range of values of
$\sigma$. After discarding the initial transients, we have
calculated the signal-to-noise-ratio measure $ \frac{<u^{2}> -<u>^{2}}{<u>}$ for $u(t,x,y)$ obtained
at $t = 3000$ and the average is taken over the entire spatial
domain \cite{pikovsky1997coherence}. The plot of the SNR measure is shown at
Fig.~\ref{fig:6} when $\sigma$ varies within the range
$[0.0025,0.1]$. From the figure, it is evident that the relative
variance of the population is not a monotonic increasing function of
the noise intensities. The environmental driving force can cause the
dispersion of the populations from their average value but the
internal mechanism can compensate to maintain the size of the
population patches and as a result we find that the relative
variance is an increasing function of the noise intensity.

\begin{figure}
\leavevmode \centering
\includegraphics[width=7.0cm]{snr_mes}
\caption{\footnotesize Signal to Noise ratio measure for $ \alpha =
2.15$, $\beta = 1$, $\gamma = 0.6$, $\delta = 0.1$, $d = 8.5$. } \label{fig:6}
\end{figure}

\section{Conclusion}

In this paper, we have studied a predator-prey model with ratio-dependent functional response and density dependent death rate of predator, incorporated with the self-diffusion terms corresponding to the random movement of the individuals within two dimensional habitats. Extensive numerical simulations were performed to understand the regimes of patterns that the model exhibited within and outside of the Turing boundary.  Typically, four types of patterns were observed within the Turing domain and interacting spiral pattern as non-Turing pattern. The effect of environmental driving forces (noise) were also studied. Interestingly, we  observed that the small amplitude noise  does not alter the stationary pattern; only the time taken to reach the stationary pattern increases. However, irregularity increases even with small noise intensities within the non-stationary regime. We observed that dispersion of population from their spatial average does not increase gradually with the noise intensities; rather noise and demographic interaction play crucial roles to determine the distribution of populations within their habitats, even during the adverse environmental conditions. Our next attempt will be to address the issues of parameter values within the spatiotemporal chaotic domain, and also investigate different noise scenarios such as  $\sigma_1\neq \sigma_2$.

\section*{References}

\bibliographystyle{iopart-num}

redpajama_set_name: RedPajamaArXiv


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