text:
\section{Introduction}
\begin{figure*}
	\centering
	\includegraphics[width=1.0\linewidth]{fig1_schematic.png}
	\caption{Illustrating game-environment feedback and the tragedy of the commons: $N_1$ number of type A individuals (`green' consumers with harvesting rate $e_L$) and $N_2$ number of type B individuals (`red' consumers with harvesting rate $e_H>e_L$) harvest $N_3$ unit of self renewing common resource (whose growth rate is $r_b$). Due to the higher harvest rate, the frequency of the type B individuals increases more than that of the type A. Although the resource keeps trying to renew itself, its overexploitation may eventually (here at time $t=t_3$) leads to the collapsing tragedy of the commons, i.e., $N_3=0$.}
	\label{fig:fig1}
\end{figure*}
Survival of an individual's lineage amid threats from the opponents in a population is dependent on its fitness that is highly influenced by the surrounding environment to which the population adapts. A favourable environmental state is beneficial for the individual and, partially, it is in the hands of the individual to keep the environment favourable. Unfortunately, the myopic selfishness of individuals involved in strategic interactions with the rest of the population may lead to overexploitation of the resources present in the environment and, thus, the environmental state is degraded to such a level that the fitness takes a hit. This degradation of the environment is commonly known as the tragedy of the commons (TOC)~\cite{lloyd1833two, hardin1968s, ostrom1999coping}. The population under question need not be human; it could even be non-human population (e.g, that of plants~\cite{Zhang2000,gersani2001tragedy,Falster2003,Day2003,zea2006tragedy}, animals~\cite{Dall2005,Rankin2006}, insects~\cite{Frank1995,Fournier2003,Wenseleers2004,Helanter2004}, microbes~\cite{Frank1996,Rainey2003,Hodgson2004,Roode2005,MacLean2006, Kerr2006}, cancer cells~\cite{Steven2004}, etc.) in which the individuals' cognitive capacity and rationality are almost next to none when compared with that of the modern humans.

In the context of the TOC in general biological evolutionary systems, the commons, i.e., the resources, are categorized into three types~\cite{Rankin2007}: extrinsic goods, social goods created by mutual cooperation, and social goods formed due to restraining from conflicts. Moreover, whether the TOC means complete or partial devastation of the resource leads to the naming of the respective tragedy as collapsing or component. This paper looks to investigate both collapsing and component tragedies of extrinsic goods exploited by the individuals evolving in line with the theory of natural selection~\cite{fisher1930book} and their feedback on the fitnesses of the individuals.

To this end, recently~\cite{weitz2016pnas,tilman2020nc} the formalism of evolutionary game theory~\cite{smith1972book,smith1973nature,smith1974jtb,smith1982book} and the deterministic replicator dynamics~\cite{taylor1978mb,schuster1983jtb,schuster1985bbpc,cressman2014pnas} have proven to be very insightful in studying the dynamics of the aforementioned feedback. However, the motivation behind all the governing equations therein is rather phenomenological. In the light of the existing literature on microscopic stochastic birth-death process~\cite{traulsen2005prl,lin2019prl,2021MCChaos} leading to replicator equation in the mean field limit and that on the microscopic process leading similarly to deterministic population dynamics equation (e.g., logistic equation~\cite{mendez2015stochastic}), in this paper we are motivated to derive the macroscopic model of the game-environment feedback dynamics starting from a microscopic multidimensional birth-death process that incorporates the feedback between the evolutionary system and an ecological self-renewing resource. Furthermore, this endeavour automatically facilitates the study of finite sized population with a finite carrying capacity which is known to have quite non-trivial counter-intuitive effects~\cite{Bairagya2021}. In passing, we remark that there do exist extensive investigations using stochastic models~\cite{Melbinger2010,Cremer2011,Melbinger2015,Wienand2017,Wienand2018} that, however, do not include an explicit equation for resource's dynamics and capture the environmental changes mostly through rather ad-hoc time-dependent carrying capacity.

After we derive the macroscopic deterministic eco-evolutionary dynamics, we focus on the surprisingly overlooked~\cite{tilman2020nc} scenario where the intrinsic growth rate of the resource is intermediate between the rates with which it is harvested by the population of selfish individuals. Such a setup is naturally suited to address both the collapsing and the component versions of the TOC; for the growth rate being lower than the lowest harvesting rate, only the collapsing TOC is expected, while for the growth rate being higher than the highest harvesting rate, only the component TOC is expected. We elaborately discuss the possible dynamics outcomes and also investigate the potential role of the finite carrying capacity of the population in averting the TOC. However, before we discuss these results below, we systematically explain and derive the mathematical model on which our entire paper is based.

\section{Deriving the model}

Let us assume that the individuals (players) of a population harvest a shared resource pool whose instantaneous state is denoted by denoted $n(t)$ that is normalized such that $0 \leq n(t) \leq 1$. Let $N_i(t)$ be the number of the individuals who are using $i$-th  action and $N(t) = \sum_{i} N_{i}(t)$ be instantaneous size of the population. Obviously, $x_i(t)=N_i(t)/N$ is the fraction of the total population using $i$th action at any instant of time $t$. The state of the harvesting population can be written as a frequency vector ${\bm x}(t) \equiv [x_1(t),\, x_2(t),\, \cdots,\, x_{\mu}(t)]$, each element of which is also interpretable as the probabilities of choosing $i$-th action by an arbitrary agent from that population.

As far as  the evolution of ${\bm x}(t)$ is concerned, within the paradigm of natural selection, the standard replicator equation is the most appropriate model. However, in this paper, we want to see the effect of finite carrying capacity in the coupled dynamics of the replicator and the resource by allowing feedback between the resource and the players. Therefore, we intend to extend the replicator equation so as to derive the set of equations that describes the time evolution of the composite state, $\sigma(t) \equiv  [n(t), {\bm x}(t), N(t)]$, of the full eco-evolutionary system described above. We would like to derive the equations starting from the microscopic birth-death process. In the simplest non-trivial setting, we choose $\mu=2$ and therefore the composite state can be alternatively and equivalently represented by ${\bm N}\equiv[N_1(t),\,N_2(t),\,N_3(t)]$ where $N_3(t)$ is unnormalised $n(t)$. The motivation behind this choice is made clear in the following discussion.

\subsection{Stochastic birth-death process}
For the sake of concreteness, without any loss of generality of the question we are interested in, consider the following scenario (see FIG.~\ref{fig:fig1}): In a consumer population of size $N$, $N_1$ number of (type A) players harvest a resource with a rate $e_L$ that is lower than the harvesting rate $e_H$ employed by the rest $N_2=N-N_1$ number of players (type B). The resource's state is characterized by $N_3$ harvestable units of C (say); e.g., fishes in a pond, trees in a forest, bacteria targeted by bacteriophages, etc. The resource is self-renewing, meaning it has a birth rate ($r_b$). Additionally, due to competition, there is a death rate ($r_d$). The effective birth rate of a harvester depends on the common resource ($N_3$) and the type of opponent it interacts with. There are death rates due to interactions between the harvesters; the death rates are assumed to be independent of the resource state that only assists in contributing to the fitness of the harvesters. The entire birth-death process can, thus, be represented as follows:
\begin{center}
    \ce{A + A  ->[$b_{11}(N_3)$] A + A + A}\\
	\ce{B + B  ->[$b_{22}(N_3)$] B + B + B}\\

    \begin{tikzpicture}
		\node (a) at (0, 0) {A+B};
		\node (b) at (2.5, 0.3){A+B+A};
		\node (c) at (2.5, -0.3){A+B+B};
		\draw (a) -- (0.7,0)--(0.7,0.3);
		\draw[->](0.7,0.3)-- (b) node[pos=0.5,above] {$\scriptstyle{b_{12}(N_3)}$};
		\draw (a)-- (0.7,0)--(0.7,-0.3);
		\draw[->](0.7,-0.3)-- (c) node[pos=0.5,above] {$\scriptstyle{b_{21}(N_3)}$};
	\end{tikzpicture}

	\ce{A + A ->[$d'_{11}$] A}\\
	\ce{B + B ->[$d'_{22}$] B}\\

    \begin{tikzpicture}
		\node (a) at (0, 0) {A+B};
		\node (b) at (1.5, 0.3){B};
		\node (c) at (1.5, -0.3){A};
		\draw (a) -- (0.7,0)--(0.7,0.3);
		\draw[->](0.7,0.3)-- (b) node[pos=0.5,above] {$\scriptstyle{d'_{12}}$};
		\draw (a)-- (0.7,0)--(0.7,-0.3);
		\draw[->](0.7,-0.3)-- (c) node[pos=0.5,above] {$\scriptstyle{d'_{21}}$};
	\end{tikzpicture}

	\ce{C ->[$r_b$] C + C}\\
	\ce{C + C ->[$r_d$] C}\\
	\ce{C  ->[$xe_L + (1-x)e_H$] $\phi$}
\end{center}

Here $b_{ij}$'s are the birth rates that depend on the resource state $N_3$ and $d'_{ij}$'s are the death rates. Also, here $x\equiv N_1/N$ and hence $1-x=N_2/N$ are the instantaneous fractions of type A and type B harvesters, respectively; therefore, there is depletion of the resource by the harvesters at a rate $xe_L + (1-x)e_H$.

Proceeding further, we can now calculate the transition probabilities per unit time. The transition  rates of type A or type B individuals to increase by one are respectively given by
\begin{equation}
T^+_1(\bm{N})={N}_1 \left[b_{11}(N_3)\frac{{N}_1-1}{{N} -1} +  b_{12}(N_3)\frac{{N}_2}{{N} -1}\right],
\end{equation}
and
\begin{equation}
T^+_2(\bm{N})={N}_2 \left[b_{21}(N_3)\frac{{N}_1}{{N} -1} +  b_{22}(N_3)\frac{{N}_2-1}{{N} -1}\right].
\end{equation}
Similarly, the transition  rates for going from $({N}_1, {N}_2,N_3)$ to $({N}_1-1, {N}_2,N_3)$ and from $({N}_1, {N}_2,N_3)$ to $({N}_1, {N}_2-1,N_3)$ respectively are
\begin{equation}
T^-_1(\bm{N})={N}_1 \left(d'_{11}\frac{{N}_1-1}{{N} -1} +  d'_{12}\frac{{N}_2}{{N} -1}\right),
\end{equation}
and
\begin{equation}
T^-_2(\bm{N})={N}_2 \left(d'_{21}\frac{{N}_1}{{N} -1} +  d'_{22}\frac{{N}_2-1}{{N} -1}\right).
\end{equation}
Furthermore, the transition rate of going from $N_3$ units of self-renewing common resource to $(N_3+1)$ units is for any state is
\begin{equation}
T_{3}^+(\bm{N})= N_3 r_b.
\end{equation}
Lastly, the probability of going from $N_3$ to $N_3-1$ per unit time is
\begin{align}
T_{3}^-(\bm{N})=\frac{N_3(N_3-1)}{2}r_d + N_3 \left(xe_L+(1-x)e_H\right),
\end{align}
where the first term on the right hand side accounts for the number of possible matched pairs leading to the possible decline in the resource, and the second term accounts for the decline due to harvesting.

Let $P(N_1,N_2,N_3,t)$ be the probability of having ${N}_1$ number of type A and ${N}_2$ number of type B individuals harvesting $N_3$ units resource at an instant of time $t$. Suppose that the carrying capacities of  type A and type B consumers, and the resource are $K_1$, $K_2$ and $K_3$ respectively. One should choose $K_1=K_2$ as the carrying capacity of the consumer population. Thus, we can normalize the variables $N_1,N_2, N_3$ as follows: $\tilde{N_i}=N_i/K_i$. The probability density function in terms of these normalized coordinates is $\rho(\tilde{N_1},\tilde{N_2},\tilde{N_2},t)=P(N_1,N_2,N_3,t)K_1K_2K_3$. The discrete master equation for the model can therefore be written as,
\begin{align}
	&\rho(\tilde{N_1},\tilde{N_2},\tilde{N_2},,t+\delta t)=\nonumber\\
	&\sum_{\alpha_1,\alpha_2,\alpha_3}\rho(\tilde{N_1}+\frac{\alpha_1}{K_1},\tilde{N_2}+\frac{\alpha_2}{K_2},\tilde{N_3}+\frac{\alpha_3}{K_3},t)\nonumber\\
	&\phantom{\sum_{\frac{\alpha_1}{K},\frac{\alpha_2}{K},\frac{\alpha_3}{k}}}\times\Gamma^{-\alpha_1}_1(\tilde{N_1}+\frac{\alpha_1}{K_1},\tilde{N_2}+\frac{\alpha_2}{K_2},\tilde{N_3}+\frac{\alpha_3}{K_3},t)\nonumber\\
	&\phantom{\sum_{\frac{\alpha_1}{K},\frac{\alpha_2}{K},\frac{\alpha_3}{k}}}\times\Gamma^{-\alpha_2}_2(\tilde{N_1}+\frac{\alpha_1}{K_1},\tilde{N_2}+\frac{\alpha_2}{K_2},\tilde{N_3}+\frac{\alpha_3}{K_3},t)\nonumber\\
	&\phantom{\sum_{\frac{\alpha_1}{K},\frac{\alpha_2}{K},\frac{\alpha_3}{k}}}\times\Gamma^{-\alpha_3}_3(\tilde{N_1}+\frac{\alpha_1}{K_1},\tilde{N_2}+\frac{\alpha_2}{K_2},\tilde{N_3}+\frac{\alpha_3}{K_3},t),
	\label{MasterA}
	\end{align}
	where $\alpha_i \in \{-1,0,1\}$, $\Gamma^{-1}_i(\tilde{\bm{N}},t)\equiv T^-_i(\tilde{\bm{N}},t)\delta t$,
  $\Gamma^{0}_i(\tilde{\bm{N}},t)\equiv(1-T^+_i(\tilde{\bm{N}},t)\delta t-T^-_i(\tilde{\bm{N}},t)\delta t)$, and $\Gamma^{1}_i(\tilde{\bm{N}},t)\equiv T^+_i(\tilde{\bm{N}},t)\delta t$ $\forall i\in\{1,2,3\}$.
 In the continuum limit, i.e., $1/K_i \to 0$ and $\delta t\to 0$, we can write the corresponding Fokker--Planck equation as follows:
\begin{widetext}
\begin{align}
\frac{\partial{\rho(\tilde{\bm{N}},t)}}{\partial{t}}=&- \sum_{i={1,2,3}} \frac{1}{K_i} \frac{\partial}{\partial{{\tilde{N}}_{i}}}\left[(T^+_{i}(\tilde{\bm{N}})-T^-_{i}(\tilde{\bm{N}}))\rho(\tilde{\bm{N}},t)\right]+\frac{1}{2}\sum_{i={1,2,3}} \frac{1}{K^2_i}  \frac{\partial^2}{\partial{{\tilde{N}}_{i}^2}}\left[(T^+_{i}(\tilde{\bm{N}})+T^-_{i}(\tilde{\bm{N}}))\rho(\tilde{\bm{N}},t)\right]\nonumber\\&+\frac{1}{2}\sum_{i={1,2,3}} \frac{1}{K_i K_j} \frac{\partial^2}{\partial{{\tilde{N}}_i}\partial{{\tilde{N}}_j}}\left[\left(T^+_{i}(\tilde{\bm{N}})-T^-_{i}(\tilde{\bm{N}})\right)\left(T^+_{j}(\tilde{\bm{N}})-T^-_{j}(\tilde{\bm{N}})\right)\rho(\tilde{\bm{N}},t)\right]\left( 1-\delta_{ij}\right).
\label{FK}
\end{align}
\end{widetext}

\subsection{Mean-field equations}
From Eq.~(\ref{FK}), using standard arguments~\cite{Gillespie1996,Vankampen}, we can write the mean field dynamics of
$\overline{N}_i\equiv K_i \int \tilde{N}_i \rho(\tilde{\bm{N}},t) d \tilde{\bm {N}} $ as
\begin{equation}
\frac{d\overline{{N}}_i}{dt}= T^+_{i}(\overline{\bm{N}})-T^-_{i}(\overline{\bm{N}}).
\label{n_i dot}
\end{equation}
Therefore, explicitly we can write the dynamics in terms of the mean total number of the individuals (${\overline{{N}}}\equiv  \overline{N}_1 +\overline{N}_2$), the mean fraction ($\overline{x}\equiv\overline{N}_1/N$) of type A individuals, and the normalization mean number of resource population ($\overline{n}\equiv\overline{N}_3/\mathcal{N}$; $\mathcal{N}$ is some normalization constant that we shall elaborate on later in the paper)  as respectively written below:
\begin{subequations}
\begin{eqnarray}
\frac{d\overline{ N}}{dt}&=&{\overline{N}} \overline{\pi}(x,\overline{N},\overline{n})\label{Ndot},\qquad\qquad\\
\frac{d{\overline x}}{dt}&=&{\overline x}(1-{\overline x})\left[\pi_1({\overline x},\overline{N},\overline{n})-\pi_2({\overline x},\overline{N},\overline{n})\right] \label{xdot},\qquad\qquad\\
\frac{d\overline{n}}{dt}&=& r_b\overline{n}\left(1-\frac{\overline{n}}{k/\mathcal{N}}\right)-\overline{n}\left[\overline{x}e_L+(1-\overline{x})e_H\right].\label{N3dot}\qquad
\end{eqnarray}
\end{subequations}
Here, with $d_{ij}\equiv d'_{ij}/N~\forall i,j\in \{1,2\}$, we have defined,
\begin{subequations}
\begin{eqnarray}
\pi_1(\overline{x},\overline{N},\overline{n})&\equiv&b_{11}(\overline{n})\overline{x}\left[1-\frac{\overline{N}}{\frac{b_{11}(\overline{n})}{d_{11}}}\right] \nonumber\\
&&+b_{12}(\overline{n})(1-\overline{x})\left[1-\frac{\overline{N}}{\frac{b_{12}(\overline{n})}{d_{12}}}\right], \\
\pi_2(\overline{x},\overline{N},\overline{n})&\equiv&b_{21}(\overline{n})\overline{x}\left[1-\frac{\overline{N}}{\frac{b_{21}(\overline{n})}{d_{21}}}\right]\nonumber\\
&&+b_{22}(\overline{n})(1-\overline{x})\left[1-\frac{\overline{N}}{\frac{b_{22}(\overline{n})}{d_{22}}}\right], \\
\overline{\pi}(\overline{x},\overline{N},\overline{n})&\equiv& \overline{x}\pi_1(\overline{x},\overline{N},\overline{n})+(1-\overline{x})\pi_2(\overline{x},\overline{N},\overline{n}),\label{r_N(x)}\qquad\qquad\\
k&\equiv&K_3=\frac{r_b}{2r_d}.
\end{eqnarray}
\end{subequations}
Having obtained the mean field equations [Eq.~(\ref{Ndot})--(\ref{N3dot})] that describes the composite deterministic state $\bm{\sigma}(t)$, we henceforth drop the overhead bar for denoting the mean quantities for the sake of notational convenience without any scope of ambiguity.

The birth rates are assumed to be modulated by the resource availability; there is an implicit ecological resource (with carrying capacity in general, dependent on $n$ and interaction between harvesters) facilitating the growth of the population. The harvesting of the common resource leads to further enhancement of the birth rates that, however, may be different for different states of the resource. The simplest forms of the birth rates could be
\begin{subequations}
\begin{eqnarray}
&b_{11}({n})=(1-{n})R_0+ {n} R_1,\\
&b_{12}({n})=(1-{n})S_0+ {n} S_1, \label{coupling}\\
&b_{21}({n})=(1-{n})T_0 + {n} T_1,\\
&b_{22}({n})=(1-{n})P_0+ {n} P_1.
\end{eqnarray}
\end{subequations}
Here, $R_0,~R_1,~S_0,~S_1,~T_0,~T_1,~P_0,$ and $P_1$ are nonnegative numbers. This immediately furnishes a game theoretic viewpoint of the situation at hand: Any arbitrarily chosen individual can have two strategies---harvesting with low rate $e_L$ and harvesting with high rate $e_H>e_L$---and the corresponding payoff matrix can be expressed as

\begin{eqnarray}
{\sf{U}}&&(n,N)=n\left[\begin{matrix}
R_1\left(1-\frac{N}{K_{11}(n)}\right)  &  S_1 \left(1-\frac{N}{K_{12}(n)}\right)\\
\\
T_1 \left(1-\frac{N}{K_{21}(n)}\right) &  P_1\left(1-\frac{N}{K_{22}(n)}\right)\\
\end{matrix}\right]\nonumber\quad\\
&&\quad+(1-n)\left[\begin{matrix}
R_0\left(1-\frac{N}{K_{11}(n)}\right)  &  S_0 \left(1-\frac{N}{K_{12}(n)}\right)\\
\\
T_0 \left(1-\frac{N}{K_{21}(n)}\right) &  P_0\left(1-\frac{N}{K_{22}(n)}\right)\\
\end{matrix}\right],\qquad
\label{pay-off matrix}
\end{eqnarray}
where $K_{ij}(n)\equiv b_{ij}(n)/d_{ij}$ ~$\forall i,j\in\{1,2\}$ are the carrying capacities. Therefore, the $i$th type (either type A or type B) individual's fitness, which depends on the state of the composite system and frequency of the types, is $f_{i}(\boldsymbol\sigma) = \sum_{j=1}^{2} {\sf U}_{ij}({\bm n},N) x_{j}.
$ Hence, the final eco-evolutionary dynamics of the composite system, that we were seeking in the beginning of this section, can be cast in the following form [see Eq.~(\ref{Ndot})--(\ref{N3dot})]:
\begin{subequations}
\begin{eqnarray}
	\frac{dN}{dt} &=& N\sum_{i=1}^{2} \sum_{j=1}^{2} {\sf U}_{ij}(n,N) x_{i}x_j,
	\label{N_eq_model}\\
\frac{dx}{dt} &=&  x\left[\sum_{j=1}^{2} {\sf U}_{1j}(n,N) x_{j} - \sum_{i=1}^{2}\sum_{j=1}^{2} {\sf U}_{ij}(n,N)  x_ix_{j}\right],\qquad\,\label{x_eq_model}\\
\frac{d{n}}{dt}&=& r_b{n}\left(1-\frac{{n}}{k/\mathcal{N}}\right)-{n}\left[{x}e_L+(1-{x})e_H\right].\label{n_eq_model}\qquad
\end{eqnarray}
\end{subequations}
\subsection{The relevant resource dynamic}
\label{parameters}
The fate of the resource, with a given carrying capacity $k$, is explicitly determined by three factors---its intrinsic growth rate and the rates at which it is harvested. Since by definition, $e_H>e_L$, following three scenarios  are exhaustive:
\begin{enumerate}
\item {$r_b<e_L<e_H$}:
This case is rather trivial as the intrinsic growth rate of the self-renewing resource is unable to sustain the resource against its constant depletion at relatively higher rates. Note that $r_b[1-n/(k/\mathcal{N})]<xe_L+(1-x)e_H$ since $x,n/k\in[0,1]$. Hence, eventually $n=0$ because $dn/dt<0$ $\forall x,n$; i.e., TOC is inevitable for such a resource. We are not concerned with this trivial case in this paper.
\item{$r_b>e_H>e_L$}: In the other extreme, if the intrinsic growth rate is higher than the high harvest rate, it is easily seen that about $n=0$ state $dn/dt>0$ leading to unconditional prevention of the collapsing TOC. Of course, the resource state cannot reach its maximum potential, $n=k$, in the presence of the harvesters. In other words, the component TOC is may still realized. To mathematically capture this scenario, it was shown~\cite{tilman2020nc} that it is convenient to choose $\mathcal{N}=(e_H-e_L)k/r_b$. Furthermore, $n=(r_b-e_H)/(e_H-e_L)$ could be choosen as the new origin so that the rescaled $n$ (at all times) is normalized to remain between zero and one. The negative (rescaled) resource state is never accessible by the construction of the model; for the same reason, it does not make sense to take any initial condition negative. Note, however, that shifting the origin to $-(r_b-e_H)/(e_H-e_L)$ is merely a convention suited to the scenario under consideration. In general, this shift is unnecessary, as is crystal clear in the next case.
\item{$e_H>r_b>e_L$}: This is arguably the most interesting case because there is a trade-off  between the low and the high harvesting rates for the resource with an intrinsic growth rate lying between the two harvesting rates. Here, collapsing TOC is present as well. Hence, it is natural to choose $\mathcal{N}=k(1-e_L/r_b)$---the maximum achievable value of the resource---and recast Eq.~(\ref{N3dot}) for this case as
\begin{equation}
\qquad\,\,\frac{1}{\varepsilon}\frac{d{n}}{dt}=r_b{n}\left[1-\left(1-\frac{e_L}{r_b}\right){n}\right]-{n}\left[{x}e_L+(1-{x})e_H\right].\label{ndot}
\end{equation}
Here, we have included a factor $\varepsilon<1$ that models the fact that the environment commonly evolves at a slower timescale than the replicators; mathematically, it merely amounts to redefining $r_b$, $e_H$, and $e_L$. Under the present normalization, the minimum resource state (that was made the origin in case 2 above) when $r_b>e_H>e_L$ is $n=(r_b-e_H)/(r_b-e_L)$ which corresponds to the component TOC. The minimum resource state, when {$e_H>r_b>e_L$}, is simply zero, which corresponds to the collapsing TOC.
\end{enumerate}
The rest of the paper deals with the eco-evolutionary game-environment feedback dynamics governed by Eq.~(\ref{N_eq_model}), Eq.~(\ref{x_eq_model}), and Eq.~(\ref{ndot}). We remark that Eq.~(\ref{ndot}) is not exclusive to case 3 (i.e., {$e_H>r_b>e_L$}) and is of most general validity; however, only for this case, the non-negativity of $n$ at all times is guaranteed.

\section{Results}
We now plan to carry out stability analysis of various possible eventualities of the composite system, and hence find out when TOC could be averted and also study the effect of the finiteness of the carrying capacities. However, first we must contrast the cases $r_b>e_H>e_L$ with $e_H>r_b>e_L$ in when all the carrying capacities $K_{ij}\to \infty~\forall i,j\in\{1,2\}$  that happens when $d_{ij} \to 0~\forall i,j\in\{1,2\}$.
\subsection{Infinite carrying capacity}

\begin{figure*}
	\centering
	\includegraphics[width=1\linewidth]{fig2_different_intrinsic_growth_rate.png}
	\caption{All possible distinct eventual outcomes for the composite state $(n,x)$ in the population with infinite carrying capacity: We vary both $\Delta^0_L$ and $\Delta^1_H$ from $-8$ to $+8$ while keeping  $\Delta^1_L$ and $\Delta^0_H$ fixed such that the four columns of subplots correspond to four exhaustive combinations of signs of $\Delta^1_L$ and $\Delta^0_H$. For the sake of concreteness and illustrative purpose, and without any loss of generality, we have fixed $\Delta^1_L$ and $\Delta^0_H$ to $-1$ and $-1$, $+1$ and $+1$, $-1$ and $+1$, and $+1$ and $-1$ respectively in the four consecutive columns.  The upper row, where $r_b>e_H>e_L$, exhibits realization of component TOC at worst; whereas the lower row, where $e_H>r_b>e_L$, depicts realization of collapsing TOC. Again, for concreteness and illustrative purpose, we have chosen $(r_b,e_H,e_L)$ to be $(1.5,1.0,0.5)$ in the former and $(1.0,1.5,0.5)$ in  the latter. Here $\varepsilon=0.2$; other values of $\varepsilon$ produce qualitatively equivalent figures.}
	\label{fig2_comparision}
\end{figure*}
In this limit, implicitly one is assuming that the consumer population has grown to its infinite capacity and is being kept fixed there. Therefore, the dynamics becomes two-dimensional because only Eq.~(\ref{x_eq_model}), and Eq.~(\ref{ndot}) are relevant. Moreover, the payoff matrix given in Eq.~(\ref{pay-off matrix}) is also simplified to
\begin{equation}
{\sf{U}}({n})={n}\left[\begin{matrix}
R_1  &  S_1 \\
T_1 &  P_1\\
\end{matrix}\right]+(1-{n})\left[\begin{matrix}
R_0  &  S_0 \\
T_0  &  P_0\\
\end{matrix}\right].
\label{pay-off matrix for infinite carrying capacity}
\end{equation}

\subsubsection{Linear stability results}
The fate of the resource and the fractions of the harvester types depend on the intrinsic growth rate of the resource and the harvesting rates, and on the payoffs. Specifically, the dependence on payoff enters through the idea of incentives: In an interaction of two individuals, incentive measures the change in the payoff of the focal player on unilaterally shifting from her current strategy to the other one. We define the following four incentives:
\begin{enumerate}
	\item  $\Delta_H^{0}\equiv S_0 - P_0$ is the incentive for low harvesting in the depleted common resource state, i.e., $n=0$, when in an interaction with a high harvester;
	\item $\Delta_L^{0}\equiv R_0 - T_0$ is the incentive for low harvesting in the depleted common resource state, i.e., $n=0$, when in an interaction with a low harvester.
	\item $\Delta_H^{1}\equiv P_1 - S_1$ is the incentive for high harvesting in the replete common resource, i.e., $n=1$, when in an interaction with a high harvester;
	\item $\Delta_L^{1}\equiv T_1 - R_1$ is the incentive for high harvesting in the replete common resource state, i.e., $n=1$, when in an interaction with a low harvester;
\end{enumerate}

In the specific parameter range of our interest, i.e., $r_b>e_L$, the fixed points, $(n^*,x^*)$, can be put in one of the following three classes: fixed points associated with TOC $(n^*=0)$, fixed points associated with complete prevention of TOC $(n^*=1)$, and the ones associated with partial prevention of TOC $(0<n^*<1)$. These are written below:
\begin{enumerate}
	\item {Realization of TOC:} There are three fixed points corresponding to this case: $(0,0)$, $(0,1)$ and  $(0,\Delta^0_H/(\Delta^0_H-\Delta^0_L))$ . The second one is always unstable. The first fixed point is stable when inequalities $r_b<e_H$ and $\Delta^0_H<0$ hold simultaneously. Whereas the third fixed point is stable when $[(r_b-e_L)\Delta^0_H-(r_b-e_H)\Delta^0_L]/(\Delta^0_H-\Delta^0_L) <0$ and $\Delta^0_H \Delta^0_L/(\Delta^0_H-\Delta^0_L)<0$ hold simultaneously.

	\item{Complete prevention of TOC:} Only one fixed point $(1,1)$ corresponds to the complete prevention of TOC. This fixed point becomes stable only when $\Delta^1_L<0$.
	%
	\item{Partial prevention of TOC:} This case has two fixed points associated with it. The first one $(n^*,x^*)=((r_b-e_H)/(r_b-e_L),0)$, which exists when $r_b>e_H$ and is stable if additionally $\left[ (e_H-e_L)\Delta^0_H-(r_b-e_H)\Delta^1_H\right] <0$. (This fixed point corresponding to the component TOC was reported as the TOC in the literature~\cite{tilman2020nc}). Less restrictive in existence is another fixed point, for which $0<x^*<1$ in addition to $0<n^*<1$, that exists whenever $r_b>e_L$. Analytically writing down its explicit expression and the condition for its stability is not possible. We have checked its existence and the condition for stability for a wide range of parameter values numerically that we have presented in FIG.~\ref{fig2_comparision}. More interestingly, this internal fixed point can give rise to a limit cycle attractor through the Hopf bifurcation with the change of parameters.
	\end{enumerate}

\subsubsection{Numerical results}
We use the hitherto gathered information to do appropriate numerics for two cases of $r_b>e_L$, viz., $r_b>e_H>e_L$ and $e_H>r_b>e_L$. First we reproduce the results~\cite{tilman2020nc} for the case $r_b>e_H>e_L$ (first row of FIG.~\ref{fig2_comparision}), however, now allowing for the initial states of the resource less than $(r_b-e_H)/(e_H-e_L)$. Subsequently, we present the new results concerning the case $e_H>r_b>e_L$ (second row of FIG.~\ref{fig2_comparision}) and compare with the former case.

What we plot in FIG.~\ref{fig2_comparision} are the all possible eventualities for the resource state and the cooperator fraction starting from arbitrary initial conditions. Mathematically, all possible eventualities are captured by the attractors, viz., stable fixed points and stable limit cycle, of the system. Obviously, the existence of these attractors depends on the specific combination of values of the four incentives. A few case studies on the real-life applications of some of the combination is presented in a recent paper~\cite{tilman2020nc}; more can be contemplated in various realistic situations. However, in this paper, our motivation is to be theoretically exhaustive rather than being focussed on discussing some specific cases.

The central difference between the two aforementioned cases is that for $r_b>e_H>e_L$, there is no collapsing TOC, unlike what is realized for $e_H>r_b>e_L$. In the former, the state of the resource lies between $(r_b-e_H)/(e_H-e_L)$ to $1$, although the fraction of the low harvesters can go to zero. Even in the exclusive presence of the high harvesters, the resource is not fully exhausted (the grey region in FIG.~\ref{fig2_comparision}(a)--(d)); this is simply because the resource's self-renewal rate is higher than the high harvesting rate. Another interesting observation is that the limit cycle attractors appear only when $\Delta_L^1>0$ in the former case, whereas in the later case, this oscillatory partial prevention of the TOC is possible even when $\Delta_L^1<0$ (the cyan region in FIG.~\ref{fig2_comparision}(g)).

In the case $r_b>e_H>e_L$, there is bistability for $\Delta_L^1<0$ (FIG.~\ref{fig2_comparision}(a) and (c)): the partial prevention of TOC coexists with complete prevention of the TOC. While such a bistability is present even in the case $e_H>r_b>e_L$ (FIG.~\ref{fig2_comparision}(g)), many more types of bistability shows up in this case: coexistence between complete TOC (with zero cooperators) and complete prevention of TOC~(FIG.~\ref{fig2_comparision}(e)), coexistence between complete TOC (with nonzero cooperators) and complete prevention of TOC~(blue region in FIG.~\ref{fig2_comparision}(g)), and coexistence between complete TOC (with zero cooperators) and partial prevention of TOC~(spotted red region in FIG.~\ref{fig2_comparision}(h)).

\subsection{Finite carrying capacity}
Having compared the case of $e_L<r_b<e_H$ with the case $r_b>e_H>e_L$, we are now well set to comprehend the effect of the finiteness of the carrying capacity in a growing population with focus on the former case which is richer in features.
\subsubsection{Modified incentives}
In contrast to the infinite population case, here, the payoffs for choosing the low or high rates of harvesting at replete or deplete environment are not solely dependent on the birth rates; it also depends on the death rates. Naturally, the incentives defined earlier must be modified. We note that the incentives decide the stability of the fixed points, so we expect the modified incentives to do the same in the case of finite carrying capacity. Hence, we introduce the modified incentives whose utility will be apparent later.

First consider $\tilde{\Delta}^0_L\equiv d_{21}(R_0/d_{11}-T_0/d_{21})=d_{21}[K_{21}(n=1)-K_{11}(n=1)]$. In words, $\tilde{\Delta}^0_L$ is the difference (scaled by $d_{21}$) between the carrying capacity at $n=0$ (hence the superscript $0$) of an individual choosing high harvesting rate and the carrying capacity of the individual alternatively choosing low harvesting rate while interacting with another individual choosing low harvesting strategy (hence the subscript $L$). If we adopt the convention that $d_{21}/d_{11}\to1$ as $d_{21},d_{11}\to0$, we note that $\tilde{\Delta}^0_L\to\Delta^0_L$ as the death rates vanish.

With similar interpretations, we furthermore define $\tilde{\Delta}^0_H$, $\tilde{\Delta}^1_L$, and $\tilde{\Delta}^1_H$ respectively as
\begin{eqnarray*}
&&d_{12}(S_0/d_{12}-P_0/d_{22})=d_{12}[K_{12}(n=0)-K_{22}(n=0)],\\
&&d_{21}(T_1/d_{21}-R_1/d_{11})=d_{21}[K_{21}(n=1)-K_{11}(n=1)],\\
&&{\rm and}\\
&&d_{12}(P_1/d_{22}-S_1/d_{12})=d_{12}[K_{22}(n=1)-K_{12}(n=1)].
\end{eqnarray*}
They all become the usual incentives in the limit of death rates going to zero.

\begin{figure*}
	\centering
	\includegraphics[width=0.9\linewidth]{fig3_topological_structure_infinite.png}
	\caption{All eventual outcomes for the composite state $(n,x)$ in the population with finite carrying capacity and $e_H>r_b>e_H$: The colour code is same as used in FIG.~\ref{fig2_comparision}; additionally, in each case, $N^*$ takes some nonzero finite values which is not needed for our purpose. We vary both $\Delta^0_L$ and $\Delta^1_H$ from $-8$ to $+8$ while keeping  $\Delta^1_L$ and $\Delta^0_H$ fixed such that the four columns of subplots correspond to four exhaustive combinations of signs of $\Delta^1_L$ and $\Delta^0_H$. For the sake of concreteness and illustrative purpose, and without any loss of generality, we have fixed $\Delta^1_L$ and $\Delta^0_H$ to $-1$ and $-1$, $+1$ and $+1$, $-1$ and $+1$, and $+1$ and $-1$ respectively in the four consecutive columns. Similarly, we vary both $\tilde\Delta^0_L$ and $\tilde\Delta^1_H$ from $-8$ to $+8$ while keeping  $\tilde\Delta^1_L$ and $\tilde\Delta^0_H$ fixed such that the four columns of subplots correspond to four exhaustive combinations of signs of $\tilde\Delta^1_L$ and $\tilde\Delta^0_H$. In order to change the signs of these two modified incentives, we use four different combinations of death rates $d_{11}$, $d_{12}$, $d_{21}$, and $d_{22}$ (first row) $d_{11}=10^{-3}$, $d_{12}=1.5\times10^{-3}$, $d_{21}=1.5\times10^{-3}$, and $d_{22}=10^{-3}$; (second row) $d_{11}=1.5\times10^{-3}$, $d_{12}=10^{-3}$, $d_{21}=10^{-3}$, and $d_{22}=1.5\times10^{-3}$; (third row) $d_{11}=10^{-3}$, $d_{12}=10^{-3}$, $d_{21}=1.5\times10^{-3}$,  and $d_{22}=1.5\times10^{-3}$; (fourth row) $d_{11}=1.5\times10^{-3}$, $~d_{12}=1.5\times10^{-3}$, $d_{21}=10^{-3}$, and $d_{22}=10^{-3}$. We fix $R_1=S_1=R_0=S_0=5.0$. For concrete illustrative purpose, we have chosen $\varepsilon=0.2$; and $r_b$, $e_H$, and $e_L$ to be $1.0$, $1.5$, and $0.5$ respectively.}
	\label{fig3_effect_of_finiteness}
\end{figure*}
\subsubsection{Linear stability results}

The first step of analysis is to study the features of the fixed point $(n^*,x^*,N^*)$ of the set of equations given by Eq.~(\ref{N_eq_model}), Eq.~(\ref{x_eq_model}), and Eq.~(\ref{ndot}). In the specific parameter range of our interest, i.e., $r_b>e_L$, there are some fixed points---viz, $(0,1,R_0/d_{11})$, $(0,0,0)$,  $(0,1,0)$, $(1,1,0)$, $((r_b-e_H)/(r_b-e_L),0,0)$, $(0,x^*\in[0,1],0)$ and $(n^*\in[0,1],x^*\in[0,1],0)$---that are always unstable and hence of no physical interest to us. As before, rest of the fixed points can be put in one of the following three exhaustive classes:
\begin{enumerate}
\item {Realization of TOC:}
The fixed point, $({n}^*,{x}^*,{N}^*)=(0,0,P_0/d_{22})$, corresponds to the TOC.  It is stable when $\tilde{\Delta}^0_H<0$ and when the higher harvesting rate is greater than the intrinsic growth of the resource, i.e., $e_H>r_b>e_L$. There is one more fixed point (with non-zero values of ${x^*}$ and ${N}^*$) which corresponds to TOC. However, its explicit closed analytical form is hard, and consequently, its stability analysis is analytically intractable as well. Hence, we are forced to undertake its analysis numerically to find when it exists and is stable, as is discussed later in this paper.

\item {Complete prevention of TOC:}
The fixed point $({n}^*,{x}^*,{N}^*)=(1,1,R_1/d_{11})$, corresponds to the full prevention of the TOC. This fixed point becomes stable when the condition $\tilde{\Delta}^1_L<0$ is satisfied.

\item {Partial prevention of TOC:}
This case has two fixed points. One is $({n}^*,{x}^*,{N}^*)=((r_b-e_H)/(r_b-e_L),0,[(e_H-e_L)P_0+(r_b-e_H)P_1]/[d_{22}(r_b-e_H)])$ that exists only when the intrinsic growth rate of the resource $r_b$ is higher than the higher harvesting rate $e_H$.  This prevention of the TOC when all the individuals are harvesting slower than the rate at which the self-renewing of the resource occurs is quite intuitive. This fixed point is stable when $(e_H-e_L)\tilde{\Delta}^0_H-(r_b-e_H)\tilde{\Delta}^1_H <0$. The other fixed point is the interior one, i.e., $x^*$ and $N^*$ are nonzero and non-unity. Again, its the analytical investigation is not conducive due to the absence of an explicit expression for it. This fixed point can undergo a change of stability through the Hopf bifurcation, and a limit cycle attractor can emerge; a stable limit cycle can also be interpreted to correspond to the partial prevention of TOC.
\end{enumerate}

\subsubsection{Numerical results}
Consider the fixed point $(0,0,P_0/d_{22})$ that corresponds to TOC. We have analytically seen that it is stable if the carrying capacity for the high rate harvesters is more than the low rate harvester in the depleted environment while interacting with another high rate harvester. This makes the low rate harvesters less effective, and eventually, all the individuals become high rate harvesters. Since the intrinsic growth rate of the resource is assumed to be less than the high harvesting rate, the TOC is inevitable in such a situation (FIG.~\ref{fig3_effect_of_finiteness}(a)--(d) and FIG.~\ref{fig3_effect_of_finiteness}(m)--(p)) as implied by the stability of the fixed point.

Similarly, the total prevention of TOC happens when the system converges to the fixed point $(1,1,R_1/d_{11})$. This fixed point becomes stable if the carrying capacity at $n=1$ for the low rate harvester is more than that of a high rate harvester while interacting with another low rate harvester. In that situation, the high rate harvesters become less effective, and all individuals eventually become low rate harvesters. Since the intrinsic growth rate of the resource is higher than the low harvesting rate, the TOC is completely averted in this case~(FIG.~\ref{fig3_effect_of_finiteness}(a)--(d) and FIG.~\ref{fig3_effect_of_finiteness}(i)--(l)).

Other than these two fixed points, the stabilities of the other fixed points, and hence the corresponding state of the resource, depend on the modified incentives in nontrivial ways. So we study the system over a wide range of parameters space, and the exhaustive results are compactly presented in FIG.~\ref{fig3_effect_of_finiteness}.

The crucial observation is that as the carrying capacities are made finite, depending on their exact values, any of the composite states can be realized for any combination of the incentives (compare any row of FIG.~\ref{fig3_effect_of_finiteness} with the second row of FIG.~\ref{fig2_comparision}) even if the state is forbidden in the case of infinite carrying capacity. It means that depending on the values of carrying capacities, the TOC can be averted when it is inevitable in an infinite population, or the TOC can be caused when it is avertable in an infinite population. Furthermore, it is crystal clear from the inspection of the trends along any column that the modified incentives are the relevant parameters in the finite carrying capacity case.

\section{Conclusions}
The common-pool resource harvesting is an interesting topic of bio-economical research~\cite{clark1976mathematical} and the eco-evolutionary dynamics that models it has an even wider application: It arguably~\cite{tilman2020nc} models the feedback between plants and soil-microbes~\cite{Bever1997} and also models the decision making~\cite{Rand2017} by agents in the co-evolving environment. It goes without saying that the assumption of an infinite fixed population is quite an unrealistic one in such cases, and hence this paper's contribution in extending the eco-evolutionary dynamics to include the effect of the growing consumer population is very pertinent.

To this end, we have derived the mean-field deterministic eco-evolutionary dynamics from a stochastic birth-death process keeping the finiteness of the populations---both the consumer and the resource---in mind. We have shown that the prevention of the TOC can be effected due to the finiteness of the consumer population through the emergence of bistability and limit cycles in the state space. More importantly, we have systematically investigated the hitherto neglected case where the resource's growth rate lies between the two distinct harvesting rates employed by the consumers. The realization and the prevention of the TOC have been found to be crucially dependent on the incentives. We have witnessed the possibility of both the component and the collapsing TOC. Furthermore, we have found that the collapsing TOC is very much dependent on the initial condition: For a given set of parameters, even if the collapsing TOC appears to be the ultimate fate, it is not so for all possible initial conditions---for the same incentives, sometimes the TOC can be completely bypassed owing to the phenomenon of bistability.

One may tweak the eco-evolutionary dynamics with a view to understanding how different strategies---e.g., reward and punishment~\cite{Mondal2022}---of averting TOC may be effective. Our setup dealing with the finite population may further be explored to study the effect of demographic noise and fluctuating environmental noise~\cite{foster1990tpb,stollmeier2018unfair,Huang2015}. Before we end, we remark that in the light of the very interesting experiments on the interplay between evolution and ecology in changing populations---e.g., in yeast~\cite{sanchez2013feedback} and bacterial~\cite{Wienand2015, Becker2018} populations---we are hopeful that some experiments with microbes~\cite{Korolev2011_1,Korolev2011_2,Frey2010,Pfeiffer2005,Li2015,Lenski2001} may be designed to validate the ideas expounded in this paper.

\acknowledgements
Researches of JDB and DC have been respectively supported by Prime Minister's Research fellowship (govt. of India) and J.C. Bose National fellowship (SERB, India). SC acknowledges support of SERB (DST, govt. of India) project MTR/2021/000119.

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Noticiero Venevisión es el noticiero de la cadena televisiva venezolana Venevisión. Anteriormente este noticiero llevaba por nombre El informador. Fue creado en 1 de marzo de 1961. Pertenece a la Alianza Informativa Latinoamericana.

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En 1973 se cambia su nombre por El informador (reservando este nombre solo para la emisión estelar, avances informativos, resumen matutino y/o boletines extraordinarios). La emisión meridiana se hacía llamar "El informador en Lo que pasa en el mundo". En 1993 pasa a llamarse simplemente El informador en todas sus emisiones.

En 1994 adquiere el nombre de Noticiero Venevisión, idea original del Noticiero Univisión.

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A partir de 2012 el noticiero forma parte de la Alianza Informativa Latinoamericana, siendo el sustituto de El Observador de RCTV. En esta alianza se establece alianzas con canales de diferentes países de América Latina, con los cuales comparte contenidos informativos, estos canales son Latina Televisión de Perú; Artear (Canal 13 y Todo Noticias) de Argentina; TV Azteca de México; Ecuavisa de Ecuador; Canal 4 de Uruguay; TVN de Panamá; Unitel de Bolivia, Telefuturo de Paraguay, y Mega de Chile. Además Noticiero Venevisión cuenta con los servicios televisivos de agencias como Reuters y AP y alianzas con canales como Univisión y CNN en Español.

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El noticiero con los años ha tenido diversos anclas principales, varios de estos teniendo una sección propia o común dentro del noticiero. En sustitución de los principales, los reporteros (principalmente los presentes en el Distrito Capital) pueden ser llegar a ser los sustitutos de los anclas en los momentos que estos no puedan estar presentes, además de poder tener también una sección en el programa. Comúnmente en la sección de espectáculos algunas de las candidatas ganadoras del Miss Venezuela llegan a conducir este espacio por unas semanas definidas.

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Raising funds for your charity or non-profit is made simpler with text messaging. Two-way messaging helps you organize your donors to raise more funds for individual projects and for general funding.
Your donors believe in you. That's why they contribute to your projects. Their enthusiasm can help you find more donors, raise more funds, and get more people in your communities excited about your work. When you make it fun for them to help, it's even better.
Every project or activity can be engaging, and fun, when you turn it into a game or contest. Text messaging helps you create that fun environment for fund-raising. You can attract and engage with donors to get them involved in your fundraising goal. Running competitions for prizes and recognition is a fun and productive way to get donors involved.
For example, you can award prizes for donors who capture different levels of investment, create campaigns within their local networks, contribute new fundraising ideas and more. You can also offer some type of annual recognition for donors who reach the highest levels of referral fund-raising, as well as recognition on a project by project basis. The options are only as limited as your imagination!
Here are some examples on how you can use text messaging to engage donors in your fund-raising opportunities.
Text messaging makes it easy to gamify fund-raising. You can announce the program, send promotional texts to keep the game top of mind, feature donors and their fund-raising ideas, offer special prizes and share each donor's personal status with them.
First, you need to decide on your offer/game strategy. Let's say you're funding a project to build a new women's center for healthcare, counseling, and support with childcare/family services.
Please note: In the conversations below, green represents an automated conversation, yellow represents a 1:1 conversation.
Following is an example message flow to announce the project, attract and engage your donors and reward them for their support.
The above messages are automatically sent as a conversation to all selected donors on your list. If you want to expand your list, it's simple to send the same message campaign at any point in time.
When donors reply to join the project, the following messages are automatically triggered.
After a week, those donors who haven't sent in a contest form get the following message. You can automatically send as many reminders as you want, at whatever time schedule you prefer.
When a donor's contest form is received, the following message flow is automatically sent.
You can also run a set of introductory events, either online or in person, to help donors raise funds, and to help them better understand the projects.
First, you'd announce the event to your donors.
When someone replies asking for help, Amy or one of her team members is immediately alerted by SMS-Magic so that they can start a 1:1 conversation.
Amy just sent a scheduling message. When Rachel schedules a call, she'll be sent a confirmation message, as well as a reminder for the call.
Text messaging gives you the chance to have important conversations with your donors, for fundraising and more.
Getting your donors involved in supporting your fund-raising efforts is a great way to deepen your relationship with them, even as you raise more funds. That's just one option to use text messaging to increase the efficiency and volume of your fund-raising initiatives.
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This British Blue bull has just been put out to do his "work" with his herd of cows to produce calves next spring.
The British Blue breed has been developed from the Belgian Blue since the 1980's and our local farmer has been instrumental in helping this process. These animals are bred for their low fat meat which fetches a premium with butchers and consumers even when crossed with non pedigree cows. Luckily for us they have good temperament and don't mind us crossing the fields on the public footpaths. This is useful for the farmers too as they need to be able to handle the bulls especially when showing them.


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Elephant Ivory Tracking
Effects of poaching on African elephants
Impacts of oil exploration
Southern resident killer whales
Northern spotted owls
Monitoring fisher populations
Grizzly bears in Jasper
Mammals of the Brazilian Cerrado
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Conservation Canines
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Center for Conservation Biology
Human population growth and consumption are placing ever-increasing demands on the environment. The Center for Conservation Biology's mission is to develop and apply noninvasive field, lab and analytical methods to address pressing conservation problems worldwide.
Wildlife monitoring
We develop and apply comprehensive tools to cost-effectively gather vast amounts of genetic, physiological and ecological data over very large terrestrial and marine habitats, along with creative ways of integrating and analyzing this information. These monitoring programs focus on a wide variety of species and frequently take advantage of our detection dog program, Conservation Canines.
CK9 Alli in Northern California.
Dogs locate scat (feces) of multiple target species over large remote areas and from considerable distances, trumping more technical tracking methods. We extract DNA, a variety of hormones, toxins and diet measures from these samples and use this information to quantify changes in the health, abundance and distribution of species threatened by one or more human disturbances. The data we generate indicate the causes of population decline, the magnitude of the problem, and what mitigation strategies are likely to be most effective.
Wildlife forensics
We apply the same approach to DNA-based, wildlife forensics on a continent-wide scale. The burgeoning illegal wildlife trade is destroying biodiversity at rates that are now rivaling those of habitat loss. Our DNA-based forensics tools are helping wildlife authorities combat the rampant illegal elephant ivory trade across Africa. This work is a collaborative effort between our center, the US Department of State, Interpol, the UN Office on Drugs and Crime, the World Bank, the CITES Secretariat, and the World Customs Organization. Wildlife authorities in many individual countries are also key players in this collaboration.
Processing seized ivory samples.
By pinpointing poaching hot spots, authorities are better able to direct law enforcement efforts; we are able to identify strategies used by organized crime syndicates driving this illegal trade; and governments are forced to take responsibility for the magnitude of the problem in their country. Our forensic techniques are readily applicable to other endangered species and we are collaborating with other scientists to develop similar tools for other species such as tigers.
Research validation
Research and development, including careful validation experiments, are essential to our mission of providing the conservation community with cost and time effective monitoring tools. Our combined field and laboratory techniques enable us to non-invasively acquire essential biological information from numerous individuals over large geographic areas. We continually expand our measures to acquire the most comprehensive estimates of wildlife population health, as well as the most accurate determinations of where major wildlife poaching events are most concentrated.
Most of our studies rely on animal scat because it is readily accessible in nature and it contains an enormous amount of information. However, sample degradation in a variable environment necessitates meticulous validation of these measures. Validation confirms the biological significance of the products being measured in scat, how these products change with time on the ground, and how these processes vary across species and environmental conditions. Validation also indicates how best to preserve samples in the field for subsequent analyses, as well as how to optimize extractions of the necessary products from the sample. All of these experiments are necessary to assure that results can withstand the many scientific and legal challenges that stem from the political and economic implications of conservation work.
Validation studies are time and cost intensive due to the need for numerous controls, multiple groups, and large sample sizes. Our Center has devoted considerable resources to such studies. We intend to continue these development and validation efforts, as we strive to expand and improve upon available monitoring tools.
Outreach & capacity building
The Center capitalizes on the excitement surrounding its innovative and applied research to foster public awareness of the challenges facing wildlife, as well as to help build capacity for conservation in developing countries.


From deep with the jungle comes a rumble and an impending sense of foreboding. A hand emerges from behind a tree, following by.. A banana.
Then with a shriek and a yell, Bouncy Monkey pops out to spread mischief and the occasional handful of faeces into your life!
Bouncy Monkey has been monkeying around since the year 2000, and was originally part of an idea to sell unique emails to you the internet connoisseur. This was when the internet craze was still heading for it's first dot com crash, and everyone thought you could sell snow to Eskimos.
Still, the monkey hung around through all this, and featured on greetings cards for friends and family, and grew in stature enough to be kidnapped and used on the websites of a local radio station, and even the BBC, until actions were taken against them (involving threats of faeces).
These days the monkey is on holiday, back in the jungle, happy back in his house that has a peculiar smell. Have a look in this here gallery to see the monkey over the years. Be sure to wear a mack..


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Home / Audiobook / History / The Romanovs Audiobook
The Romanovs Audiobook
The Oil Kings Audiobook $39.95 $4.99
The United States of Absurdity Audiobook $20.95 $4.99
by Robert K. Massie
Narrated by: Geoffrey Howard
In July 1991, nine skeletons were exhumed from a shallow mass grave near Ekaterinburg, Siberia, a few miles from the infamous cellar room where the last tsar and his family had been murdered 73 years before. But were these the bones of the Romanovs? And if these were their remains, where were the bones of the two younger Romanovs supposedly murdered with the rest of the family? Was Anna Anderson, celebrated for more than 60 years in newspapers, books, and film, really Grand Duchess Anastasia? The Romanovs provides the answers, describing in suspenseful detail the dramatic efforts to discover the truth.
Pulitzer Prize winner Robert K. Massie presents a colorful panorama of contemporary characters, illuminating the major scientific dispute between Russian experts and a team of Americans, whose findings, along with those of DNA scientists from Russia, America, and Great Britain, all contributed to solving one of the great mysteries of the 20th century.
Robert K. Massie
Geoffrey Howard
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A discografia do grupo de eurodance alemão Culture Beat consiste em quatro álbuns de estúdio, um álbum de remixes, duas coletâneas e vinte e três singles.

O grupo foi formado em 1989 e lançou seu álbum de estreia em 1991, mas só conquistou o reconhecimento internacional com o lançamento do segunda álbum de estúdio, Serenity, lançado em 1993 e que vendeu mais de 1,5 milhões de cópias.

Do álbum foi extraído o maior sucesso do grupo, o single "Mr. Vain",  número 1 em 13 países, incluindo Alemanha, Austrália e Reino Unido.

Álbuns

Álbuns de estúdio

Álbuns de compilação

Álbuns de remixes

Singles

Referências

Culture Beat


I am 28 and married. I have brown eyes and hair. I am tall and fit. I have been studying English since 2007. Since then I am trying to build up my vocabulary but I am not a hardworker student. I always lose my control, whenever I make a study plan I always break the rules. So I am a person who always need a guide to push me study. Unfortunately I have no self-control. I am quite determined these days to build up my voc, hopefully this website can help me. I want to take IELTS exam this is why I want to learn more words. I can easilyunderstand what I read but when it comes to writing I can not remember a single Word, it is the same with speaking. However my speaking is better than my writing. This is my story.
When I pronounce this word I really feel awesome. The way it sound is just great. I think people who do not know its meaning and who come across this word for the first time can easily can understand its meaning.
I always keep forgetting its spelling, I mostly misspell it. It is a very simple Word and I learned it at the begnning when I strated learnig English but it makes lose my selfconfident. As it makes me think even I can not remember how to write this basic Word how I can build up my vocabulary.