On a prey-predator reaction-diffusion system with Holling type III functional response

In this paper we study a prey-predator model defined by an initial-boundary value problem whose dynamics is described by a Holling type III functional response. We establish global existence and uniqueness of the strong solution. We prove that if the initial data are positive and satisfy a certain regularity condition, the solution of the problem is positive and bounded on the domain Q=(0,T)x@W and then we deduce the continuous dependence on the initial data. A numerical approximation of the system is carried out with a spectral method coupled with the fourth-order Runge-Kutta time solver. The biological relevance of the comparative numerical results is also presented.

[1]  V. Křivan,et al.  Alternative Food, Switching Predators, and the Persistence of Predator‐Prey Systems , 2001, The American Naturalist.

[2]  Marcus R. Garvie.  Finite-Difference Schemes for Reaction–Diffusion Equations Modeling Predator–Prey Interactions in MATLAB , 2007, Bulletin of mathematical biology.

[3]  Ralf Seppelt,et al.  "It was an artefact not the result": A note on systems dynamic model development tools , 2005, Environ. Model. Softw..

[4]  V. Barbu.  Mathematical Methods in Optimization of Differential Systems , 1994 .

[5]  Marcus R. Garvie,et al.  Numerische Mathematik Finite element approximation of spatially extended predator – prey interactions with the Holling type II functional response , 2007 .

[6]  C. S. Holling,et al.  The functional response of predators to prey density and its role in mimicry and population regulation. , 1965 .

[7]  Wan-Tong Li,et al.  Multiple bifurcations in a predator-prey system with monotonic functional response , 2006, Appl. Math. Comput..

[8]  Yinnian He,et al.  Traveling wavefronts for a two-species ratio-dependent predator–prey system with diffusion terms and stage structure , 2009 .

[9]  Lloyd N. Trefethen,et al.  Fourth-Order Time-Stepping for Stiff PDEs , 2005, SIAM J. Sci. Comput..

[10]  Shenghua Xu.  EXISTENCE OF GLOBAL SOLUTIONS FOR A PREDATOR-PREY MODEL WITH CROSS-DIFFUSION , 2008 .

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

[12]  Wan-Tong Li,et al.  Traveling waves in a diffusive predator–prey model with holling type-III functional response , 2008 .

[13]  M. Scheffer.  Ecology of Shallow Lakes , 1997, Population and Community Biology Series.

[14]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[15]  H. De Meyer,et al.  Exponentially fitted Runge-Kutta methods , 2000 .

[16]  Yihong Du,et al.  A diffusive predator–prey model with a protection zone☆ , 2006 .

[17]  S. Krogstad.  Generalized integrating factor methods for stiff PDEs , 2005 .

[18]  Marten Scheffer,et al.  Fish and nutrients interplay determines algal biomass : a minimal model , 1991 .

[19]  R. Nisbet,et al.  Response of equilibrium states to spatial environmental heterogeneity in advective systems. , 2006, Mathematical biosciences and engineering : MBE.

[20]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[21]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[22]  S. Cox,et al.  Exponential Time Differencing for Stiff Systems , 2002 .

[23]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[24]  Satish C. Reddy,et al.  A MATLAB differentiation matrix suite , 2000, TOMS.

[25]  M. Langlais,et al.  Some remarks on a singular reaction-diffusion system arising in predator-prey modeling , 2007 .