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Zbl 0834.92019
Calsina, Àngel; Perelló, Carles; Saldaña, Joan
Non-local reaction-diffusion equations modelling predator-prey coevolution. (English)
[J] Publ. Mat., Barc. 38, No.2, 315-325 (1994). ISSN 0214-1493

A prey-predator system with a characteristic of the predator subject to mutation is studied. The considered model is $$u_t = \left( \varphi (u) - \int^1_0 hv \right) u, \quad v_t = \bigl( x + h(x)u - \mu \bigr) v + dv_{xx}, \tag 1$$ where $u(t)$ represents the prey-- population and $v(x,t)$, $x \in [0,1]$, represents the predator population. The function $v$ satisfies the Dirichlet boundary conditions $v(0,t) = v(1,t) = 0$. The function $\varphi$ is a logistic term for the growth rate and the coefficients $d,h$ and $\mu$ have specific interpretations. \par The authors conclude an evolutionary stable strategy (EES) result for the diffusion coefficient tending to zero (Th. 3.1). It is also proved that there exists $d_0 > 0$ such that, for all $d > d_0$, there exists an equilibrium solution of the system (1) (Th. 4.1).
[I.Onciulescu (Iaşi)]

MSC 2000:: *92D25 Population dynamics
35K57 Reaction-diffusion equations
92D15 Problems related to evolution

Keywords: prey-predator system; mutation; Dirichlet boundary conditions; evolutionary stable strategy; equilibrium solution

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