Lee, Sungrim Seirin

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TitleThe Effect of the Remains of Carcass in Two-prey, one-predator model with Diffusion(Kyoto Winter School ofMathematical Biology 2006)Author(s)Lee, Sungrim SeirinCitation数理解析研究所講究録 (2007), 1556: 131-137Issue Date2007-05URL http://hdl.handle.net/2433/81000RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityThe Effect of the Remains of Carcass in Two-prey,one-predator model with Diffusion(Sungrim Seirin Lee)Graduate School of Environmental Science,Okayama University1 IntroductionIn nature, there are many various and complex interactions between creatureson several steps of trophic level. Many studies related with such a population in-teraction have done and shown coexistence and various dynamics of populations.As one of the related studies, continuous time models for food chain, exploitativecompetition, apparent competition and intraguild predation have been researchedextensively in literature (e.g.[l], [3] and so on). As an example of the interactionwhich is not described above, we find out that an omnivorous cricket eats vegetablesas well as the carcass of some small animals. Usually, no all of predators eat a wholeprey and thus some parts of the carcass of the prey remain. Then the remains ofcarcass can be foods for the other animals.For convenience, henceforth the omnivorous prey will be called the existent preyand the small animal, the invader prey. Then, the following system gives the simplestmodel which is concerned with the effect of the remains of carcass in two-prey, onepredator mode1([2]).$\frac{d}{dt}h_{1}=\epsilon_{1}(1-\frac{h_{1}}{k_{1}})h_{1}-a_{1}h_{1}p+\tau_{1}h_{2}ph_{1}$$\frac{d}{dt}h_{2}=\epsilon_{2}(1-\frac{h_{2}}{k_{2}})h_{2}-a_{2}h_{2}p$ (1)$\frac{d}{dt}p=-\delta p+b_{1}h_{1}p+b_{2}h_{2}p$where $h_{1},$ $h_{2}$ and $p$ denote the densities of the existent prey, the invader prey andthe predator respectively. In the model (1) the amount of the remains of carcass issimply proportional to the rate that the predator meet the invader prey. To intendthe effect of the carcass, we suppose that the existent prey and the invader prey donot have inter-specific competition and the other direct interactions.Now, in this paper we consider a diffusion model which is concerned with thenegative effect by intra-specific competition of existent preys and obstruction of1556 2007 131-137 131invader preys. The functional response of the remains of carcass with negativeeffect is given as the following:$\Psi(h_{1}, h_{2},p)=\frac{\tau_{1}h_{2}p}{\tau_{2}h_{1}+\tau_{3}h_{2}+1}$.In this paper, we notice the case that $\tau_{1},$ $\tau_{2}$ and $\tau_{3}$ are sufficiently large. By sim-plifying the functional response $\Psi(h_{1}, h_{2},p)$ mathematically with $\tau_{1}/\tau_{3}=\mu_{1}$ and$\tau_{2}/\tau_{3}=\mu_{2}$ as $\tau_{3}arrow\infty$ , we obtain the following model:$\frac{\partial}{\partial t}h_{1}=d_{1}\frac{\partial^{2}}{\partial x^{2}}h_{1}+\epsilon_{1}(1-\frac{h_{1}}{k_{1}})h_{1}-a_{1}h_{1}p+\frac{\mu_{1}h_{2}p}{\mu_{2}h_{1}+h_{2}}h_{1}$$\frac{\partial}{\partial t}h_{2}=d_{2}\frac{\partial^{2}}{\partial x^{2}}h_{2}+\epsilon_{2}(1-\frac{h_{2}}{k_{2}})h_{2}-a_{2}h_{2}p$ (2)$\frac{\partial}{\partial t}p=d_{3}\frac{\partial^{2}}{\partial x^{2}}p-\delta p+b_{1}h_{1}p+b_{2}h_{2}p$.Then we have the following nondimensionlized system without loss of generality.$\frac{\partial}{\partial t}h_{1}=d_{1}\frac{\partial^{2}}{\partial x^{2}}h_{1}+(1-h_{1})h_{1}-\alpha_{1}h_{1}p+\frac{\omega h_{2}p}{kh_{1}+h_{2}}h_{1}$$\frac{\partial}{\partial t}h_{2}=d_{2}\frac{\partial^{2}}{\partial x^{2}}h_{2}+\epsilon(1-h_{2})h_{2}-\alpha_{2}h_{2}p$ (3)$\frac{\partial}{\partial t}p=d_{3}\frac{\partial^{2}}{\partial x^{2}}p-\gamma p+\beta_{1}h_{1}p+\beta_{2}h_{2}p$ .where$\alpha_{i}=\frac{a_{1}}{\epsilon_{1}}$ , $\omega=\frac{\mu_{1}}{\epsilon_{1}}$ , $k=\mu_{2^{\frac{k_{1}}{k_{2}’}}}$ $\epsilon=\frac{\epsilon_{2}}{\epsilon_{1}}$ , $\gamma=\frac{\delta}{\epsilon_{1}}$ and $\beta_{i}=\frac{b_{1}k_{1}}{\epsilon_{1}}$ . (4)We suppose that the environment is uniform, and the existent prey and the $pred*$tor have already attained the coexistence state in the space uniformly when somepropagules of the invader prey (say $n_{0}$ individuals) invade in the origin at $t=0$ .This gives the initial condition as the following:$y_{1}(x, 0)= \frac{\gamma}{\beta_{1}}$ , $y_{2}(x, 0)=n_{0}\hat{\delta}(x)$ , $p(x, 0)= \frac{1}{\alpha_{1}}(1-\frac{\gamma}{\beta_{1}})$ (5)where $\hat{\delta}(x)$ is delta function.In this paper, we show that there exist a traveling wave by numerical simulationand propose the speed of it by mathematical analysis including numerical simula-tions. We obtain the speed condition which depends only on the diffusion coefficientof the invader. We also find out a chaotic phenomenon.2 Numerical simulation resultsIn the system (3), we have a stationary solution$E^{0}=( \overline{h}_{1}(x), 0,\overline{p}(x))=(\frac{\gamma}{\beta_{1}},0,$ $\frac{1}{\alpha_{1}}(1-\frac{\gamma}{\beta_{1}}))$132if $\beta_{1}>\gamma 1s$ satisfied. Furthermore, the statIonary solution $E^{0}$ becomes unstablewhen$\epsilon-\frac{\alpha_{2}}{\alpha_{1}}(1-\frac{\gamma}{\beta_{1}})>0$ .We also have the following result([2]).Proposition 1 There exists a positive stationary solution $E^{*}=(h_{1}^{*}(x), h_{2}^{*}(x),p^{*}(x))$uniquely if the following conditions are satisfied:$\epsilon-\frac{\alpha_{2}}{\alpha_{1}}(1-\frac{\gamma}{\beta_{1}})>0$ , $1+ \frac{\epsilon}{\alpha_{2}}(\omega-\alpha_{1})(1-\frac{\gamma}{\beta_{2}})>0$ . (6)Now we carry out numerical simulations with parameter values satisfying theconditions (6). Then we find out a traveling wave with $d_{1}=d_{2}=d_{3}=1$ and $\epsilon=1$ ,which connects $E^{0}$ and $E^{*}$ at $xarrow\infty$ and $xarrow-\infty$ respectively (Fig.1). When wetake $\epsilon=3$ , we see chaotic dynamics for some fixed $x(Fig.2)$ .Figure 1: Traveling wave with $\epsilon=1$ Figure 2: Chaos with $\epsilon=3$ for timeWe have $c\mathfrak{N}ried$ out these simulations with all the same diffusion values, butwe show that the traveling wave speed depends on only $d_{2}$ through mathematicalapproach and numerical simulation in the following section.3 Speed of traveling waveRom the numerical simulation results we have found a traveling wave solution whichconnects the stationary solution $E^{0}=(\overline{h}_{1}(x), 0,\overline{p}(x))$ and $E^{*}=(h_{1}^{*}(x), h_{2}^{*}(x),p^{*}(x))$ .Now we propose the speed of traveling wave only depends on diffusion coefficient $d_{2}$of the invader prey.We seek traveling wave solutions of system(3) in the form$U(z)=(h_{1}(x, t),$ $h_{2}(x, t),p(x, t))=(U_{1}(z), U_{2}(z),$ $U_{3}(z))$ (7)where $z=x-ct$ and $c>0$ . Then these solutions should satisfy$U(-\infty)=(h_{1}^{*}, h_{2}^{*},p^{*})$ and $U(\infty)=(\overline{h}_{1},0,\overline{p})$ . (8)133With setting $V_{1}(z)=U_{1}’(z),$ $V_{2}(z)=U_{2}’(z)$ and $V_{3}(z)=U_{3}’(z)$ , substituting (7) intosystem (2) gives the following dynamical system:$U_{1}’=V_{1}$$U_{2}’=V_{2}$$U_{3}’=V_{3}$$V_{1}’=- \frac{1}{d_{1}}\{cV_{1}+(1-U_{1})U_{1}-\alpha_{1}U_{1}U_{3}+\frac{\omega U_{2}U_{3}}{kU_{1}+U_{2}}U_{1}\}$ (9)$V_{2}’=- \frac{1}{d_{2}}\{cV_{2}+\epsilon(1-U_{2})U_{2}-\alpha_{2}U_{2}U_{3}\}$$V_{3}’=- \frac{1}{d_{3}}\{cV_{3}-\gamma U_{3}+\beta_{1}U_{1}U_{3}+oeU_{2}U_{3}\}$.Thus from the conditions (8), we also have the following conditions:$\xi^{*}=(U_{1}(-\infty), U_{2}(-\infty),$ $U_{3}(-\infty),$ $V_{1}(-\infty),$ $V_{2}(-\infty),$ $V_{3}(-\infty))$$=(h_{1}^{*}, h_{\dot{2}},p^{l}, 0,0,0)$ ,$\overline{\xi}=(U_{1}(\infty), U_{2}(\infty),$ $U_{3}(\infty),$ $V_{1}(\infty),$ $V_{2}(\infty),$ $V_{3}(\infty))$$=(\overline{h}_{1},0,\overline{p},0,0,0)$ .We can check easily that $\xi^{*}=(h_{1}^{*}, h_{2}^{*},p^{*},0,0,0)$ and $\overline{\xi}=(\overline{h}_{1},0,\overline{p}, 0,0,0)$ are equilib-ria of the dynamical system (9).Now, we find the solution of the dynamical system (9) which connects the twoequilibrium points $\xi^{*}$ and $\overline{\xi}$ and also satisfies $U_{1}(z)>0,$ $U_{2}(z)>0$ and $U_{3}(z)>0$ .We notice the positiveness of the solution $U_{2}$ because $U_{2}(z)arrow 0$ as $zarrow\infty$ . Thatis, if the solution $U_{2}$ oscillates, it must have negative values around origin. Then $U_{2}$cannot lead to traveling wave solution since the traveling wave should be a positivesolution of the system (3). To show that $U_{2}(z)$ does not oscillate around the origin,we inv\’etigate the behaviors of $U_{2}$ around the origin by linearization of the system(9)about $\overline{\xi}$.Let $J(\overline{\xi})$ denote Jacobian matrix of the dynamical system(9) at $\overline{\xi}$. Then theeigenvalues $\lambda$ of Jacobian $J(\overline{\xi})$ are given by the solutions of the following eigenvalueequation:$\det|\lambda I-J(\overline{\xi})|=\{\lambda^{2}+\frac{c}{d_{2}}\lambda+\frac{1}{d_{2}}(\epsilon-\alpha_{2}\overline{p})\}$$\{\lambda^{4}+c(\frac{d_{1}+d_{3}}{d_{1}d_{3}})\lambda^{3}+(\frac{c^{2}}{d_{1}d_{3}}-\frac{\overline{h}_{1}}{d_{1}})\lambda^{2}-\frac{c\overline{h}_{1}}{d_{1}d_{3}}+\alpha_{1}\beta_{1}\frac{\overline{h}_{1}\overline{p}}{d_{1}d_{3}}\}=0$ .where $I$ is the identity matrix. Set the first term and the second term of theeigenvalue eqation as $\phi_{1}(\lambda)$ and $\phi_{2}(\lambda)$ respectively:$\phi_{1}(\lambda)=\lambda^{2}+\frac{c}{d_{2}}\lambda+\frac{1}{d_{2}}(\epsilon-\alpha_{2}\overline{p})$$\phi_{2}(\lambda)=\lambda^{4}+c(\frac{d_{1}+d_{3}}{d_{1}d_{3}})\lambda^{S}+(\frac{c^{2}}{d_{1}d_{3}}-\frac{\overline{h}_{1}}{d_{1}})\lambda^{2}-\frac{c\overline{h}_{1}}{d_{1}d_{3}}+\alpha_{1}\beta_{1^{\frac{\overline{h}_{1}\overline{p}}{d_{1}d_{3}}}}$134Then for the eigenvalue $\lambda$ satisfying $\phi_{1}(\lambda)=0$ or $\phi_{2}(\lambda)=0$ , we can obtain theeigenvector $X$ from the following equation:$(\lambda I-J(\overline{\xi}))X=0$ , (10)where $X^{t}=(U_{1}, U_{2}, U_{3}, V_{1}, V_{2}, V_{3})$ .We first consider the eigenvalues included in $\{\lambda : \phi_{1}(\lambda)\neq 0, \phi_{2}(\lambda)=0\}$ . Thenfrom the equation (10) we obtain the following system about $U_{2}$ and $V_{2}$ :$\{\begin{array}{l}\frac{\epsilon-\alpha_{2}\overline{p}}{d_{2}}U_{2}+(\lambda+\frac{c}{d_{2}})V_{2}=0\lambda U_{2}-V_{2}=0\end{array}$Solving the system give the following equation:$\{\lambda^{2}+\frac{c}{d_{2}}\lambda+\frac{1}{d_{2}}(\epsilon-\alpha_{2}\overline{p})\}U_{2}=0$,and thus we have$U_{2}=0$ ,since $\phi_{1}(\lambda)\neq 0$ . Therefore, we conclude that $U_{2}$ does not oscillate around the originin the case of $\lambda\in\{\lambda:\phi_{1}(\lambda)\neq 0, \phi_{2}(\lambda)=0\}$ .Now we consider the remained case of the eigenvalues in $\{\lambda : \phi_{1}(\lambda)=0\}$ . Thensimilarly from eigenvector equation (10) we have$\{\begin{array}{l}-\lambda(\lambda+\frac{c}{d_{2}}U_{2})+(\lambda+\frac{c}{d_{2}})V_{2}=0\lambda U_{2}-V_{2}=0\end{array}$Solving the system gives$\lambda U_{2}=V_{2}$ ,and thus we obtain $U_{2}(z)$ ffom $V_{2}=U_{2}’$ :$U_{2}(z)=e^{\lambda z}\zeta$ ,where $\zeta$ is a constant. If $\lambda$ is not real number but complex number, $U_{2}$ must beoscillate around the origin. It cannot lead to traveling wave solution. Hence,$\emptyset i(\lambda)=\lambda^{2}+\frac{c}{d_{2}}\lambda+\frac{1}{d_{2}}(\epsilon-\alpha_{2}\overline{p})=0$should have only real solutions. For the existence of only real solutions, the followinginequality is obtained:$( \frac{c}{d_{2}})^{2}-4\frac{\epsilon-\alpha_{2}\overline{p}}{d_{2}}\geq 0$.At last we lead the condition that the speed of the traveling wave should satisfy:$c\geq 2\sqrt{d_{2}(\epsilon-\alpha_{2}\overline{p})}$ , (11)135Figure 3: Slopes of linae give the travel-ing wave speed with $d_{2}=0.5(red)$ and $d_{2}=$$2(blue)$ . In the case of $d_{2}=0.5$ we have thespeed $c=1$ .where $\overline{p}=\frac{1}{\alpha_{1}}(1_{\dot{\beta}_{1}}-l)$ . Note that we obtain this condition for arbitrary $d_{1},$ $d_{2},$ $d_{3}$ .It is well-known conjecture that the real traveling wave speed correspond tothe minimun speed of (11). That is, the traveling wave speed $c$ will be equal to2 $\sqrt{d_{2}(\epsilon-\alpha_{2}\overline{p})}$ . In fact, we examine the speed by using numerical simulations andcheck the speed of traveling wave which is equal to 2 $\sqrt{d_{2}(\epsilon-\alpha_{2}\overline{p})}$ . For instance,we carry out numerical simulations with $d_{2}=0.5$ and $d_{2}=2(Fig.3)$ and the otherparameter values as the following:$\epsilon=1$ $\alpha_{1}=\alpha_{2}=2$ , $\beta_{1}=2$ , $\gamma=1$ .Figure3 show that the speed in the case of $d_{2}=0.5$(red line), $c=\Delta x/\Delta t\approx 1$ , andthe speed in the case of $d_{2}=2$ (blue dots) is twice of the case $d_{2}=0.5$ .Therefore, we have the traveling wave solution which moves with the speed depending on the diffusion coefficient $d_{2}$ of invader prey but not $d_{1}$ and $d_{3}$ . As one ofnumerical simulation results, Figure 4 show that the speed $c$ is proportional to $\sqrt{2}$but constant to $d_{3}$ .Figure 4: Dependence of diffusion coefficients $d_{2}$ and $d_{3}$1364 ConclusionsIn this paper, we have shown that three species are able to coexist and diffusewith constant speed. Moreover, the diffusion speed depends only on the diffusioncoefficient of the invader prey. The invasion of the invader prey lead the predator’spopulation to increase. Furthermore, the remains of carcass of the invader prey alsochange the existent prey’s population. We can suppose that the effect of the invaderprey’s population is very large at the head of traveling wave. Moreover, we haveseen chaos in the diffusion model (3). The diffusion effect may give more complexfluctuation of populations.Until now, many prey-predator population models have been studied but usually,they have not considerd the existence of carcass. However, it is very important thatthe remains of carcass of some species effects the population of outsider preys aswell as themselves. In this paper, we have only considered the one predator case butwe should consider the case of more than one predator for the remains of carcass.Moreover, we should also study about the effect of remains of carcass with differentfunctional responses.References[1] Holt R.D., Polis G.A., A thmretical hamework for intraguild predation. TheAmerican Naturalist $149(1997)745- 764$[2] Lee S.S., Kajiwara T., The effect of the remains of carcass in two-prey, one-predator system, in preparation.[3] Tanabe K., Namba T., Omnivory creates chaos in simple food web models.Ecology, 86(2005) 3411-3414137

The Effect of the Remains of Carcass in Two-prey, one-predator model with Diffusion(Kyoto Winter School of Mathematical Biology 2006)

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The Effect of the Remains of Carcass in Two-prey, one-predator model with Diffusion(Kyoto Winter School of Mathematical Biology 2006)

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