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Non-local reaction-diffusion equations modelling predator-prey coevolution. (English) Zbl 0834.92019

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.
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).

MSC:

92D25 Population dynamics (general)
35K57 Reaction-diffusion equations
92D15 Problems related to evolution
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