- Volume 30 Issue 3
In the field of population dynamics and chemical reaction the possibility or the existence of spatially and temporally nonhomogeneous solutions is a very important problem. For last 50 years or so there have been many results on the pattern formation of chemical reaction systems studying reaction systems with or without diffusions to explain instabilities and nonhomogeneous states arising in biological situations. In this paper we study time-dependent properties of a predator-prey system with functional response and give sufficient conditions that guarantee the existence of stable limit cycles.
prey-predator system;functional response;diffusion;convergence property;existence of positive constant steady-state;existence of stable limit cycle
- R. Bhattacharyya, B. Mukhopadhyay and M. Bandyopadhyay Diffusion-driven stability analysis of a prey-predator system with Holling type-IV functional response, Systems Analysis Modelling Simulation, 43(2003), No. 8, 1085-1093. https://doi.org/10.1080/0232929031000150409
- A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetica 12 (1972), 30-39. https://doi.org/10.1007/BF00289234
- A. Gierer and H. Meinhardt, Application of a theory of boilogicol pattern based on lateral inhibition, J. Cell. Sci. 15 (1974), 321-376.
- S. A. Levin and L. A. Segel, Pattern generation in space and aspect, SIAM Rev. 21 (1985), 45-67.
- R. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900-902. https://doi.org/10.1126/science.177.4052.900
- J. D. Murray, Lectures on Nonlinear Differential Equation Models in Biology, Oxford University Press (1977).
- J. D. Murray, Mathematical biology, Springer-Verlag, Heidelberg (1989).
- A. Okubo and L.A. Levin, Diffusion and Ecological Problems : modern perspective, Interdisciplinary Applied Mathematics, 2nd ed., Vol. 14, Springer, New York, 2001.
- R. May, Paradox of enrichment in competitive systems, Ecology, 55 (1971), 183-187.
- M. Rosenzweig, Paradox of enrichment: destabilization of exploitation ecosystems in ecological time, Science, 171 (1971), 385-387. https://doi.org/10.1126/science.171.3969.385
- M. Rosenzweig, Reply to McAllister et al., Science, 175 (1972), 564-565.
- S.-A. Rosenzweig, Reply to Gilpin, Science, 177 (1972), 904. https://doi.org/10.1126/science.177.4052.904
- S.-A. Shim, Long-time properties of prey-predator system with cross-diffusion, Commun. Korean Math. Soc. 21 (2006), No. 2, 293-320. https://doi.org/10.4134/CKMS.2006.21.2.293
- S.-A. Shim, Global existence of solutions to the prey-predator system with a single cross-diffusion, Bull. Korean Math. Soc. 43 (2006), No. 2, 443-459. https://doi.org/10.4134/BKMS.2006.43.2.443
- A. M. Turing, Chemical basis of morphogenesis, Phil. Trans. R. Soc. London B 237 (1952), 37-72. https://doi.org/10.1098/rstb.1952.0012