A prey‐predator model with diffusion and a supplementary resource for the prey in a two‐patch environment¹

This work partially was carried out at Department of Mathematics, Indian Institute of Technology, Kanpur‐208016, India. Author is thankful to Prof. J. B. Shukla for his valuable suggestion.

Abstract

In this paper, a prey‐predator dynamics, where the predator species partially depends upon the prey species, in a two patch habitat with diffusion and there is a non‐diffusing additional resource for the prey population, is modeled and analyzed. It is shown, that there exists a positive, monotonic, continuous steady state solution with continuous matching at the interface for both the species separately. Further, we obtain conditions for asymptotic stability for both linear and nonlinear cases.

Šiame straipsnyje modeliuojama ir analizuojama plešr^‐unu ir auku dinamika, laikant, kad plešr^‐unu populiacija dalinai priklauso nuo auku skačiaus. Areala sudaro dvi sritys, kuriose vyksta populiaciju individu difuzija, be to, aukoms yra išskirtas nedifunduojantis resursas.

Irodyta, kad egzistuoja teigiamas, monotoniškas, tolydus stacionarusis sprendinys, tenkinantis tolydumo salyga abiems populiacijoms atskirai. Gautos asimptotinio stabilumo salygos tiesiniu ir netiesiniu atvejais.

Key words:

• Population diffusion
• patchiness
• supplementary resource
• steady state solution
• stability

Notes

This work partially was carried out at Department of Mathematics, Indian Institute of Technology, Kanpur‐208016, India. Author is thankful to Prof. J. B. Shukla for his valuable suggestion.