Abstract
We study a diffusive predator-prey system with a ratio-dependent functional response when a prey population is infected under homogeneous Neumann boundary condition. All non-negative and positive equilibria are investigated, and the conditions that give rise to asymptotic behavior of these equilibria are examined. In particular, we present a biological interpretation of disease-free and total extinction states. A comparison principle and the stability analysis for the parabolic problem are employed.
Highlights
We focus on the diffusive predator-prey system with a ratio-dependent functional response and disease in the prey; ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨vutt d d u = u[r r K uαw mw+u+v bv], v v[bu d βw mw+u+v ],⎪⎪⎪⎪⎪⎪⎪⎪⎩uw∂∂uη(t –=, xD∂∂)ηv==wu∂ ∂=wη(xw=),[ –d o+vn(m ( w,cx,α+∞)uu+=v)v+× (m∂xw)c,+β,uv+v ]in (, ∞) ×, w(, x) = w (x) in ( . )where ⊆ RN is a bounded region with smooth boundary ∂, and r, m, K, b, di, Di, c, α, and β are positive constants; a, b, b, l and k are positive constants as well
3 Conclusion A diffusive predator-prey model with a ratio-dependent functional response and infected prey population was investigated under homogeneous Neumann boundary conditions
We showed that depending on initial data, all species can become extinct if the predation rate is small and the searching efficiency constant of the predation rate of the predator for the susceptible prey is large; in other words, the predator overeats the susceptible prey
Summary
We investigate the conditions of the asymptotic behavior of a unique positive constant solution and the non-negative equilibria of We point out that the corresponding non-spatial model had the same asymptotic behavior under the same condition in the following theorem. The following theorem indicates that one can control the infected prey, namely, only the infected prey can be removed out under some conditions (Figure ).
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