A diffusive predator–prey model with a protection zone☆

In this paper we study the effects of a protection zone Ω0 for the prey on a diffusive predator–prey model with Holling type II response and no-flux boundary condition. We show the existence of a critical patch size described by the principal eigenvalue λ D (Ω0) of the Laplacian operator over Ω0 with homogeneous Dirichlet boundary conditions. If the protection zone is over the critical patch size, i.e., if λ D (Ω0) is less than the prey growth rate, then the dynamics of the model is fundamentally changed from the usual predator–prey dynamics; in such a case, the prey population persists regardless of the growth rate of its predator, and if the predator is strong, then the two populations stabilize at a unique coexistence state. If the protection zone is below the critical patch size, then the dynamics of the model is qualitatively similar to the case without protection zone, but the chances of survival of the prey species increase with the size of the protection zone, as generally expected. Our mathematical approach is based on bifurcation theory, topological degree theory, the comparison principles for elliptic and parabolic equations, and various elliptic estimates.

[1]  E. N. Dancer.  On positive solutions of some pairs of differential equations , 1984 .

[2]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations: Cantrell/Diffusion , 2004 .

[3]  Rui Peng,et al.  Positive steady states of the Holling–Tanner prey–predator model with diffusion , 2005, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[4]  Sze-Bi Hsu,et al.  A diffusive predator–prey model in heterogeneous environment , 2004 .

[5]  E. N. Dancer.  On the existence and uniqueness of positive solutions for competing species models with diffusion , 1991 .

[6]  M. Crandall,et al.  Bifurcation from simple eigenvalues , 1971 .

[7]  F. Rothe,et al.  Convergence to Homogeneous Equilibrium State for Generalized Volterra–Lotka Systems with Diffusion , 1979 .

[8]  Yuan Lou,et al.  Some uniqueness and exact multiplicity results for a predator-prey model , 1997 .

[9]  C. Cosner,et al.  Stable Coexistence States in the Volterra–Lotka Competition Model with Diffusion , 1984 .

[10]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[11]  Junping Shi,et al.  On a singular nonlinear semilinear elliptic problem , 1998, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[12]  Robert Stephen Cantrell,et al.  Permanence in ecological systems with spatial heterogeneity , 1993, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[13]  Charles ReVelle,et al.  Using mathematical optimization models to design nature reserves , 2004 .

[14]  Hiroshi Matano,et al.  Pattern Formation in Competition-Diffusion Systems in Nonconvex Domains , 1983 .

[15]  SPATIAL PATTERNS FOR POPULATION MODELS IN A HETEROGENEOUS ENVIRONMENT , 2004 .

[16]  A general monotone scheme for elliptic systems with applications to ecological models , 1986 .

[17]  Yihong Du.  Chapter 3 – Bifurcation and Related Topics in Elliptic Problems , 2005 .

[18]  Chris Cosner,et al.  Diffusive logistic equations with indefinite weights: population models in disrupted environments , 1991, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[19]  C. S. Holling.  Some Characteristics of Simple Types of Predation and Parasitism , 1959, The Canadian Entomologist.

[20]  Yihong Du.  Effects of a Degeneracy in the Competition Model Part II.: Perturbation and Dynamical Behaviour , 2002 .

[21]  Josep Blat,et al.  Global bifurcation of positive solutions in some systems of elliptic equations , 1986 .

[22]  Yihong Du,et al.  Allee effect and bistability in a spatially heterogeneous predator-prey model , 2007 .

[23]  Yihong Du,et al.  Effects of Certain Degeneracies in the Predator-Prey Model , 2002, SIAM J. Math. Anal..

[24]  J. G. Skellam.  Random dispersal in theoretical populations , 1951, Biometrika.

[25]  On the spectrum of some linear noncooperative elliptic systems with radial symmetry , 1995 .

[26]  Junping Shi,et al.  Optimal Spatial Harvesting Strategy and Symmetry-Breaking , 2008 .

[27]  Yuan Lou,et al.  Qualitative behaviour of positive solutions of a predator—prey model: effects of saturation , 2001, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[28]  Yihong Du,et al.  Competing Species Equations with Diffusion, Large Interactions, and Jumping Nonlinearities , 1994 .

[29]  Zongming Guo,et al.  The degenerate logistic model and a singularly mixed boundary blow-up problem , 2005 .

[30]  Yihong Du,et al.  Logistic Type Equations on RN by a Squeezing Method Involving Boundary Blow‐Up Solutions , 2001 .

[31]  C. Roberts,et al.  Effects of marine reserves on adjacent fisheries. , 2001, Science.

[32]  Konstantin Mischaikow,et al.  Convergence in competition models with small diffusion coefficients , 2005 .

[33]  K. Schmidt.  'No-Take' Zones Spark Fisheries Debate , 1997, Science.

[34]  Yihong Du.  Realization of prescribed patterns in the competition model , 2003 .

[35]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[36]  E. N. Dancer,et al.  On positive solutions of some pairs of differential equations, II , 1985 .

[37]  Julián López-Gómez,et al.  Existence and uniqueness of coexistence states for the predator-prey model with diffusion: the scalar case , 1993 .

[38]  Yuan Lou,et al.  S-Shaped Global Bifurcation Curve and Hopf Bifurcation of Positive Solutions to a Predator–Prey Model , 1998 .

[39]  Konstantin Mischaikow,et al.  Competing Species near a Degenerate Limit , 2003, SIAM J. Math. Anal..

[40]  Yihong Du,et al.  Effects of a Degeneracy in the Competition Model , 2002 .

[41]  Mingxin Wang,et al.  Qualitative analysis of a ratio-dependent predator–prey system with diffusion , 2003, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[42]  Julián López-Gómez,et al.  Existence and uniqueness of coexistence states for a predator-prey model with diffusion , 1994 .

[43]  Junping Shi,et al.  New exact multiplicity results with an application to a population model , 2001, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[44]  Junping Shi,et al.  Persistence and Bifurcation of Degenerate Solutions , 1999 .

[45]  Dag Standal.  Fishing the last frontier—controversies in the regulations of shrimp trawling in the high Arctic , 2003 .

[46]  L. Payne.  Isoperimetric Inequalities and Their Applications , 1967 .