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\begin{document}

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\Title{Synchronous Oscillatory Solutions in a Two Patch Predator-Prey Model}
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\LHead{\huge{Matthew H. Becker$^1$, Leah Shaw$^2$, Junping Shi$^1$}
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\textsl{\\ 
1. Department of Mathematics, The College of William \& Mary, Williamsburg, Virginia, 23187-8795\\ 2. Department of Applied Science, The College of William \& Mary, Williamsburg, Virginia, 23187-8795}
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\Head{Abstract}

\slshape\
"Populations of adversarial species in nature engaged in predator-prey dynamics may oscillate over time. This is seen best in the example of the lynx and hare. When this occurs, unique patches of species may synchronize such that populations in each patch are equivalent.  Oscillatory solutions to a predator-prey model are studied to understand what leads to this synchrony. Here we use Alan Hastings' version of the Rosenzweig-MacArthur model.  A correlation coefficient similar to Pearson's Correlation is used as a statistical method to quantify synchrony and we use this as a numerical tool to analyze our data and results."

 



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\Head{One Patch Model}


\begin{eqnarray*}
\frac{du}{dt}&=& u(1-\alpha u)-\frac{uv}{1+\beta u}, \\
\frac{dv}{dt}&=& \frac{uv}{1+\beta u}-\eta v
\end{eqnarray*}
%In this model, a nondimensionalized form of the Rosenzweig-MacArthur model from Alan %Hasting's paper in The Ecology Letters (2001), $u$ is the populations of the prey, with $v$ as %the population of the predator.  $K$ is the carrying capacity of the prey and $m$ is the %saturation rate of the predator.  Thus, $\frac{muv}{1+u}$ measures the amount of prey eaten by %the predators when they interact.  Additionally, $e$ measures the natural mortality rate of the %predator.

This system presents multiple equilibria, but the one we care most about is our coexistence equilibrium $(\lambda, v_\lambda)$ for $\lambda=\frac{\eta}{1-\beta\eta}$ and $v_\lambda=(1-\alpha\lambda)(1+\beta\lambda)$.  We linearize the system and study the stability using the Jacobian matrix, J,  with the characteristic equation $\mu^2-Tr(J)\mu+Det(J)=0$ for the eigenvalues of the matrix.  Through algebraic manipulation we can find the Hopf Bifurcation and see that when $0<\lambda<\frac{\beta-\alpha}{2\alpha\beta}$ a periodic orbit will occur. This is the case that we care about for when we go to a two-patch model.

\begin{figure}[t]
\includegraphics[height=3in]{bostononepatchmodeltimeseries.eps}
\end{figure}



The first plot shows the time series for unstable coexistence in a one patch model, where both populations demonstrate oscillations.  The second plots shows the phase space, where predator is mapped against prey and the periodic orbit can be seen.

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\begin{figure}[t]
\includegraphics[height=3in]{bostononepatchphasespace.eps}
\end{figure}
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\Head{Two-Patch Model}
\slshape

We use Alan Hasting's nondimensionalized coupled two-patch model from The Ecology Letters (2001).  The additional terms, with $a$ and $b$ as diffusion rates, measure the dispersal between the two patches.

\begin{eqnarray*}
\frac{du}{dt} &=& f_1(u,v)+a(w-u), \\
\frac{dv}{dt} &=& g_1(u,v)+b(x-v), \\
\frac{dw}{dt} &=& f_2(w,x)-a(w-u), \\
\frac{dx}{dt} &=& g_2(w,x)-b(x-v), 
\end{eqnarray*}
where for $i=1,2$,
\begin{equation}
    f_i(u,v)=u\left(1-\alpha_i u\right)-\frac{uv}{1+\beta_i u}, \;\;
    g_i(u,v)=\frac{uv}{1+\beta_i u}-\eta_i v.
\end{equation}





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\Head{Numerical Analysis}

\slshape
%We calculate correlation values keeping all parameters constant except diffusion rates $a$ and %$b$, which we keep as $a=3b$.  This gives us a view of how synchrony changes due to %diffusion rates.  We see from the plots below that as the diffusion rates get higher, then level of %synchrony approaches $1$ on the correlation scale.

The four sets of plots below show data collected. We assume spatial homogeneity and fix parameters $\alpha=0.01, \beta=0.04$, $\eta=1$, and $T=1000$.  We change diffusion parameters $a$ and $b$.  Initial values are $u(0)= 4, v(0)=3, w(0)=0, x(0)=0$. The far left plot is a time series of predator and prey versus time, the middle plot is the predator-prey phase space, and the far right plot is the time shift $\tau$ versus cross correlation.

\begin{figure}[h]
\begin{center}
\includegraphics[width=0.285\textwidth,keepaspectratio]{01timeseries0001.eps}
\includegraphics[width=0.285\textwidth,keepaspectratio]{01phasespace0001.eps}
%\end{figure}
%\begin{figure}[h]
\includegraphics[width=0.4\textwidth,keepaspectratio]{01ccshaw0001.eps}
%\caption{Numerical simulation of synchrony. Here $a=0.1$ and $b=0.001$. A. (upper left): Time %Series (Predator and Prey vs. time); B. (upper right): Phase portrait; C. (lower): Cross Correlation %vs. Time Shift $\tau$.  \label{fig10}}
%\end{center}
%\end{figure}
\\ Here $a=0.1, b=0.001. cc=0.9983, cc(\tau)=-3.9401\times10^{-4}$\\
%\begin{figure}
%\begin{center}
\includegraphics[width=0.27\textwidth,keepaspectratio]{001timeseries001.eps}
\includegraphics[width=0.27\textwidth,keepaspectratio]{001phasespace001.eps}
%\end{figure}
%\begin{figure}[h]
\includegraphics[width=0.4\textwidth,keepaspectratio]{001ccshaw001.eps}
%\caption{Numerical simulation of synchrony. Here $a=0.1$ and $b=0.1$. A. (Left): Time Series %(Predator and Prey vs. time); B. (upper right): Phase portrait; C. (lower): Cross Correlation vs. %Time Shift $\tau$.  \label{fig10}}
%\end{center}
%\end{figure}
\\Here $a=0.01, b=0.01. cc=0.2917, cc(\tau)=0-0.0046.$\\
%\begin{figure}[t]
%\begin{center}
\includegraphics[height=3.6in,keepaspectratio]{0001timeseries001.eps}
\includegraphics[width=0.285\textwidth,keepaspectratio]{0001phasespace001.eps}
%\end{figure}
%\begin{figure}[h]
\includegraphics[width=0.4\textwidth,keepaspectratio]{0001ccshaw001.eps}
%\caption{Numerical simulation of synchrony. Here $a=0.1$ and $b=0.01$. A. (upper left): Time %Series (Predator and Prey vs. time); B. (upper right): Phase portrait; C. (lower): Cross Correlation %vs. Time Shift $\tau$.  \label{fig12}}
%\end{center}
%\end{figure}
\\ Here $a=0.1, b=0.01. cc=0.4633 , cc(\tau)= $0.0037 \\
%\begin{figure}[t]
%\begin{center}
\includegraphics[width=0.35\textwidth,keepaspectratio]{0001timeseries0001.eps}
\includegraphics[width=0.265\textwidth,keepaspectratio]{0001phasespace0001.eps}
%\end{figure}
%\begin{figure}[h]
\includegraphics[width=0.375\textwidth,keepaspectratio]{0001ccshaw0001.eps}
%\caption{Numerical simulation of synchrony. Here $a=0.001$ and $b=0.001$. A. (upper left): %Time Series (Predator and Prey vs. time); B. (upper right): Phase portrait; C. (lower): Cross %Correlation vs. Time Shift $\tau$.  \label{fig11}}
\\Here $a=0.001, b=0.001. cc=-0.2533 , cc(\tau)=6.6797\times10^{-4}$ \\
\end{center}
\end{figure}



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\Head{Conclusions} 

\slshape
As the diffusion rates increase, the synchrony of the system also increases, although not necessarily in a linear manner.  Additionally, the cross correlation has appears to have a lower bound, situated around -0.3. The system does not achieve antisynchrony.  This is corroborated  when time is shifted in one patch, where again there is no antisynchrony present.  The time shift shows us that there are sets of parameters that will never allow for synchrony (fig3), while others that are simply out of phase (fig4). Finally, we see that the average cross correlation over a number of different time shifts, $cc(\tau)$, remains just around $0$. 


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\Head{Cross Correlation}

In order to analyze synchrony in the two patches we use a Cross Correlation model.  This model is very similar to Pearson's Correlation, one of the most widely used correlation measures in statistical analysis. It is defined$ cc(u,w; T)=\frac{\langle u(t)-\bar{u}, w(t)-\bar{w} \rangle}{\sigma_u  \sigma_w}$.

The two functions $u$ and $w$ are the prey solutions in the time span $[0,T]$ for $T\in \mathbb{R}$  to the predator-prey model above.  $\bar{u}$ and $\bar{w}$ are defined as the average value for each function over $[0,T]$, with $\bar{u}=T^{-1}\int_{0}^{T} u(t)dt$.  The numerator is akin to a vector inner product on $L^2(0,T)$, which is
$\langle f, g \rangle=\int_0^T f(t) \cdot g(t) dt.$
$\sigma_u$ is the standard deviation of  function $u$, and it is defined as follows: $\sigma_{u}=\sqrt{\int_{0}^{T}u^{2}(t)dt}$.

This equation gives a result between $-1$ and $1$, with $1$ meaning perfect synchrony, $0$ as asynchrony, and $-1$ as anti-synchrony.

Cross-Correlation values give a good indication of the synchrony of a solution at a specific time, but can not decipher whether a system might be locked out of phase.  In order to analyze more we apply an arbitrary time shift to the correlation. This is defined as $cc_\tau(u,w; T)=\frac{\langle u(t)-\bar{u}, w(t)-\bar{w}_\tau \rangle}{\sigma_u  \sigma_w}$ \\
with the numerator being the new vector inner product $\langle f, g \rangle_\tau=\int_\tau^T f(t) \cdot g(t-\tau) dt.$

\begin{figure}
\begin{center}
\includegraphics[width=0.3\textwidth,keepaspectratio]{scatter2d.eps}
\includegraphics[width=0.3\textwidth,keepaspectratio]{CorrelationPlot.eps}
\caption{Left: Correlation coefficients plotted against predator diffusion $a$ and prey diffusion $b$. Here $\alpha_1=\alpha_2=0.01$, $\eta_1=\eta_2=1$, $\beta_1=\beta_2=0.04$, $T=1000$, and $0.0001\le a\le 0.1$, $0.0001\le b\le 0.1$.  Initial values are $u(0)= 4, v(0)=3, w(0)=0, x(0)=0$. Right: Plot of the cross correlation vs. diffusion coefficients. Initial values are $u(0)= 0.8, v(0)=0.6, w(0)=0.6, x(0)=0.8$. \label{fig7}}
\end{center}
\end{figure}




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%\Head{Conclusions} 

%\slshape

%Synchronization is more likely to be achieved when diffusion is large.  Cross Correlation is never less than -0.2. Time shifted could show periodic pattern or more random pattern, which does not seem related to synchronization. The average cross correlation (with time shift) value is always right around zero.

%We see that, as might have been expected, the levels of synchrony rise as the diffusion rates get %higher.  However, there is not a linear correlation.  The cross correlation value rises very slowly %until the diffusion rates reach a certain point and then jump up to the $0.8-0.9$ area.  There %appears to be a point at which there is a swift switch from relative asynchrony to an almost %perfectly synchronous state.

%The level of synchrony will also depend upon the initial conditions.  If the initial values of the %two predator or prey populations are close enough then that will lead to synchrony independent %of the diffusion rates.  In the same way, if the initial values are so far from each other then even %with large diffusion rates the system may never reach synchrony.

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\Head{Future Work and References} 

\slshape
We will continue to work with the correlation coefficient to further understand what leads to synchronous solutions.  We have begun analyzing the Floquet multipliers to study the stability of periodic orbits. We are going to combine this with our continued study of synchrony.


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\bibliographystyle{amsalpha}
\begin{thebibliography}{99}


\bibitem{A} Glass, L.; 
Synchronization and Rhythmic Processes in Physiology.
\textsl{Nature}, \textbf{410} (2001), 277-284.

\bibitem{B} Hastings, A.;
Transient Dynamics and Persistence of Ecological Systems.
\textsl{Ecology Letters} \textbf{4} (2001), 215-220.

%\bibitem{C} Goldwyn, E. E.; Hastings, A.,
%When can dispersal synchronize populations?
%\textsl{Theo. Popu. Biol.}  \textbf{73} (2008), 395--402.

\bibitem{C} Rosenzweig, M.L. and MacArthur, R.H.;
Graphical Representation and Stability Conditions of Predator-Prey Interactions.
\textsl{American Naturalist} (1963), 209-223.

\bibitem{D}  Rodgers, J. L.; Nicewander, W. A., 
Thirteen Ways to Look at the Correlation Coefficient.  
\textsl{The American Statistician}, \textbf{42} (1988), 59-66.


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\Head{Acknowledgments} 

\slshape

This research is supported by NSF CSUMS Grant DMS-0703532 and NSF Grant DMS-1022648.

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