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Coexistence and permanence in a four-species food chain model. (English) Zbl 0837.92027

The authors consider the system \[ u_t- D_1 \Delta u= D_1 u(a_1- u+k_{12} v+k_{13} w), \quad v_t- d_2 \Delta v= D_2 v(a_2- k_{21} u-v+ k_{23} w+ k_{24} z), \tag{1} \]
\[ w_t- d_3 \Delta w= D_3 w(a_3- k_{31} u-k_{32} v-w+ k_{34} z), \quad z_t- D_4\Delta z= D_4 z(a_4- k_{42} v- k_{43} w- z) \] in \((0,\infty)\times \Omega\), \(u=v= w=z=0\) on \((0,\infty)\), \[ u(0,x)= u_0, \quad v(0,x)= v_0, \quad w(0,x)= w_0, \quad z(0,x)= z_0, \] where \(u\), \(v\), \(w\), \(z\) represent the densities of the species \(A\), \(B\), \(C\), \(D\) in a bounded environment \(\Omega\) in \(\mathbb{R}^n\) with smooth boundary \(\partial \Omega\). The species \(A\) consumes species \(B\) and \(C\), species \(B\) consumes species \(C\) and \(D\) and species \(C\) consumes species \(D\). The diffusion coefficients \(D_i\) and the intrinsic growth rates \(a_i\) \((1\leq i\leq 4)\) are assumed positive and the interaction rates \(k_{ij}\) \((1\leq i,j\leq 4)\) are all non-negative. The initial functions \(u_0\), \(v_0\), \(w_0\), \(z_0\) are assumed to be smooth, nonnegative and not identically zero in \(\Omega\).
A sufficient condition is given for the existence of a coexistence state (a componentwise positive solution \((U,V,W,Z)\) of the stationary system associated to system (1)). Under the same conditions the long-term survival for all of the species in the food chain model is also ensured. The system (1) is solved numerically on the domain \(\Omega= (0,1)\).

MSC:

92D40 Ecology
35K55 Nonlinear parabolic equations
35K57 Reaction-diffusion equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
65N99 Numerical methods for partial differential equations, boundary value problems
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