Coexistence of a one-prey two-predators model with ratio-dependent functional responses

Abstract We examine the positive coexistence of one prey and two competing predators in an interacting system with ratio-dependent functional responses under a hostile environment. Furthermore, we examine the extinction conditions for all three interacting species and for one or two of the three species. In addition, we present the biological interpretation and simulations based on the result. The methods employed are a comparison argument for the elliptic problem and the fixed-point theory applied to a positive cone on a Banach space.

[1]  C. Pao.  Coexistence and stability of a competition—diffusion system in population dynamics , 1981 .

[2]  Wonlyul Ko,et al.  A diffusive one-prey and two-competing-predator system with a ratio-dependent functional response: I, long time behavior and stability of equilibria , 2013 .

[3]  Wonlyul Ko,et al.  A diffusive one-prey and two-competing-predator system with a ratio-dependent functional response: II stationary pattern formation , 2013 .

[4]  Existence of positive solutions for a semilinear elliptic system , 2011 .

[5]  Sze-Bi Hsu,et al.  A ratio-dependent food chain model and its applications to biological control. , 2003, Mathematical biosciences.

[6]  Xin Lu,et al.  Some coexistence and extinction results for a $3$-species ecological system , 1995 .

[7]  Shigui Ruan,et al.  Intraspecific interference and consumer-resource dynamics , 2004 .

[8]  Wei Feng,et al.  Coexistence, Stability, and Limiting Behavior in a One-Predator-Two-Prey Model , 1993 .

[9]  H. Amann.  Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces , 1976 .

[10]  Nela Lakos,et al.  Existence of steady-state solutions for a one-predator-two prey system , 1990 .

[11]  Wonlyul Ko,et al.  Analysis of ratio-dependent food chain model , 2007 .

[12]  Rui Peng,et al.  Stationary Pattern of a Ratio-Dependent Food Chain Model with Diffusion , 2007, SIAM J. Appl. Math..

[13]  Lige Li,et al.  Coexistence theorems of steady states for predator-prey interacting systems , 1988 .

[14]  Yihong Du.  Positive periodic solutions of a competitor-competitor-mutualist model , 1996 .

[15]  A. Leung.  A study of three species prey-predator reaction-diffusions by monotone schemes☆ , 1984 .

[16]  Inkyung Ahn,et al.  Positive solutions for ratio-dependent predator–prey interaction systems , 2005 .

[17]  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.

[18]  Michael Pedersen,et al.  Stability in a diffusive food-chain model with Michaelis–Menten functional response , 2004 .

[19]  Sze-Bi Hsu,et al.  Rich dynamics of a ratio-dependent one-prey two-predators model , 2001, Journal of mathematical biology.

[20]  E. N. Dancer.  On the indices of fixed points of mappings in cones and applications , 1983 .

[21]  Kimun Ryu,et al.  Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics , 2003 .

[22]  S. Hsu,et al.  Global analysis of the Michaelis–Menten-type ratio-dependent predator-prey system , 2001, Journal of mathematical biology.

[23]  Kwang Ik Kim,et al.  Coexistence in the three species predator-prey model with diffusion , 2003, Appl. Math. Comput..

[24]  Yang Kuang,et al.  Global qualitative analysis of a ratio-dependent predator–prey system , 1998 .

[25]  Yihong Du,et al.  Positive solutions for a three-species competition system with diffusion—I. General existence results , 1995 .

[26]  Yaping Liu.  Positive solutions to general elliptic systems , 1995 .