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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 =φ(u)- 0 1 hvu,v t =x + h ( x ) u - μv+dv xx ,(1)

where u(t) represents the prey– population and v(x,t), x[0,1], represents the predator population. The function v satisfies the Dirichlet boundary conditions v(0,t)=v(1,t)=0. The function φ is a logistic term for the growth rate and the coefficients d,h and μ 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:
92D25Population dynamics (general)
35K57Reaction-diffusion equations
92D15Problems related to evolution