the predator-prey discrete system codimention- 2 bifurcations

Authors
raana moghadasi
reza khoshsiar
shahrekorduniversity
Abstract
Introduction

In population dynamics, discrete-time dynamical systems have been used to describe interaction between ecological species. Comparing to continuous-time dynamical systems, discrete-time models are more suitable to describe populations with non overlapping generations. These models in general produce rich and complex dynamical behaviors.

Among various population interaction, predator-prey models play a fundamental rule in mathematical ecology. The dynamics of predator-prey system is greatly depend on the implementation of the functional response, the availability of prey for predation. In this paper we consider a planar system which describes a predator-prey model. In order to reveal comprehensive dynamics of the system, we employee theoretical tools such as center manifold theorem along with numerical tools based on numerical continuation method.

Material and methods

Our analysis is based on theoretical and numerical techniques. We first determine all fixed points of the system and conditions under which these points may undergo different bifurcations. To reveal more dynamics of the system, we also use numerical bifurcation methods and numerical simulations, which further examine the obtained analytical results.

Results and discussion

For the resented discrete-time predator-prey system, we compute several bifurcation curves, all possible codimension-1 and codimension-2 bifurcations on thses curves along with their corresponding normal form coefficients. By branch switching technique and employing software package MatcontM, we compute stability boundaries for several cycles up to period 32. We also use numerical simulation, to compute basin of attraction and strange attractor emerging around a Neimark-Sacker bifurcation.

Conclusion

We can highlight the following results from this paper.

Detection and location of all fixed points of a discrete-time predator pray system.
Computing all possible codimension-1 and -2 bifurcation and their corresponding normal form coefficients which in turn reveal criticality of the bifurcation points and determine if extra bifurcation curves can emanate from each detected bifurcation.
Computing orbits up to period 32 which determine stability thresholds for different cycles.

Computing basin of attraction and strange of attractor which emerge around a Neimark-Sacker bifurcation by means of numerical simulation technique.
Keywords
Fixed point
Bifurcation
Stability
Numerical Simulation
1. Albert CLJ. "Regularity and complexuty in dynamical systems", New york: Springers 2012.

2. Broer H. W., Saleh K., Naudot V, and Roussarie R., "Dynamics of a predator-prey model with nonmonotonic response function", Discrete and Continous Dynamical System. Series A, Vol. 18, no. 2-3.

3. Danca M., Codreanu S., and Bako B., "Detailed analysis of a nonlinear prey-predator model", Journal of Biological Physics, vol. 23, no. 1, pp. 11-21, 1997.

4. Doedel E. J., Champeys R. A., Farigrieve T.F, Kuznestov Yu A., Standstede B., and Wang X. J. " AUTO2000: Continuation and Bufurcation sotware for Ordinary Differential Equations (with HomCont), Users’ Guide", Concordia University, Monteral, Canada 1997-2000

5. Dhooge, A., Govaerts W., Kuznetov Yu. A., Mestorm, W., and Riet, A, "MATCONT: A Continuation toolbox in MATLAB in Proceeding of the 2 Symposium on Applied Computing", Melborn, FL, Association for Machnery, New York, 2003, 161-166.

6. Govaerts W., Ghaziani R. K., Kunestov Yu. A., and Meijer H. G E., "Numerical methods for two parameters local bifurcation analysis of maps", SIAM Journal on Scientifuc Computing, vol. 29, no.6.

7. Govaerts W., Ghaziani R. K., "Numerical bifurcation analysis of a nonlinear stage structure cannivlism population model. J.Differentioanl Equations", Appl 12 (10)(2006) 1069-1085

8 .Govaerts W., Ghaziani R. K., Kunestov Yu. A,, Meijeir H., "Bifurcation analysis of periodic orbits of map in Matlab". 2006, prepoint

9. MATLAB, "The Mathworks Inc". http://www.mathworks.com

10. Murakami K., "Stability and bifurcation in discrete-time predator-prey model,” Journal of Difference Equation and Applications", vol. 13, no. 10, pp. 911-925, 2007.

11. Murray J. D., "Mathematical Biology, vol. 19 of Biomathematics, Springer, Berlin, Germany", 2nd edition, 1993.

12. Khan A. Q, "Neimark- Sacker bifurcation of a two-dimensional discrete-time predator-prey model", SpringerPlus (2016)5:126.

13. Kuznetsov Yu. A., and Meijer H. G. E., "Numerical normal forms for codim 2 bifurcations offixed points with at most two critical eigenvalues", SIAM Journal on Scientific Computing, 26,10.1137/030601508 .
Mathematical Research
Volume 8, Issue 2 - Serial Number 21
Spring 2022
Pages 216-241