On the effects of spatial heterogeneity on the persistence of interacting species

Abstract. The dynamics of two interacting theoretical populations inhabiting a heterogeneous environment are modelled by a system of two weakly coupled reaction–diffusion equations having spatially dependent reaction terms. Longterm persistence of both populations is guaranteed by an invasibility condition, which is itself expressed via the signs of certain eigenvalues of related linear elliptic operators with spatially dependent lowest order coefficients. The effects of change in these coefficients upon the eigenvalues are here exploited to study the effects of spatial heterogeneity on the persistence of interacting species through two particular ecological topics of interest. The first concerns when the location of favorable hunting grounds within the overall environment does or does not affect the success of a predator in predator–prey models, while the second concerns cases of competition models in which the outcome of competition in a spatially varying environment differs from that which would be expected in a spatially homogeneous environment.

[1]  Song-Sun Lin,et al.  On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function , 1980 .

[2]  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.

[3]  Josep Blat,et al.  Bifurcation of steady-state solutions in predator-prey and competition systems , 1984, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

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

[5]  Chris Cosner,et al.  Ecological models, permanence and spatial heterogeneity , 1996 .

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

[7]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[8]  M. Gilpin,et al.  Interference competition and niche theory. , 1974, Proceedings of the National Academy of Sciences of the United States of America.

[9]  Jonathan Roughgarden,et al.  Spatial heterogeneity and interspecific competition , 1982 .

[10]  J. M. Ball,et al.  GEOMETRIC THEORY OF SEMILINEAR PARABOLIC EQUATIONS (Lecture Notes in Mathematics, 840) , 1982 .

[11]  Stefan Senn,et al.  On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions, with an application to population genetics , 1983 .

[12]  C. Bandle.  Isoperimetric inequalities and applications , 1980 .

[13]  P. Yodzis,et al.  Introduction to Theoretical Ecology , 1989 .

[14]  Andrew Sih,et al.  Prey refuges and predator prey stability , 1987 .

[15]  Chris Cosner,et al.  Should a Park Be an Island? , 1993, SIAM J. Appl. Math..