Persistence and stability for a two-species ratio-dependent predator-prey system with distributed time delay. (English) Zbl 1008.34065

Xu, R.; Davidson, F. A.; Chaplain, M. A. J.

J. Math. Anal. Appl. 269, No. 1, 256-277 (2002).

The family of models introduced here and the results achieved are generalisations to a wider class of delays of the ones obtained in [E. Beretta and Y. Kuang, Nonlinear Anal., Theory Methods Appl. 32, No. 3, 381-408 (1998; Zbl 0946.34061)]. The reference system without delays is the Michaelis-Menten predator-prey model which is very similar to the better known Michaelis-Menten-Holling predator-prey model.
For this family of models, the uniform persistence of positive solutions is proven under certain conditions. The unique positive equilibrium of this system is then considered and results on its local and global stability properties are presented. An example of this generalised family of models is then introduced and explored numerically with MATLAB.

Reviewer: Domingo Salazar (Oxford)

MSC:

34K20	Stability theory of functional-differential equations
92D25	Population dynamics (general)

Keywords:

time delay; uniform persistence; local stability; global stability; Michaelis-Menten predator-prey model

Citations:

Zbl 0946.34061

Software:

Matlab

References:

[1]	Arditi, R.; Saiah, H., Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73, 1544-1551, (1992)
[2]	Arditi, R.; Ginzburg, L.R.; Akcakaya, H.R., Variation in plankton densities among lakes: a case for ratio-dependent models, Amer. nat., 138, 1287-1296, (1991)
[3]	Arditi, R.; Perrin, N.; Saiah, H., Functional response and heterogeneities: an experiment test with cladocerans, Oikos, 60, 69-75, (1991)
[4]	Gutierrez, A.P., The physiological basis of ratio-dependent predator – prey theory: a metabolic pool model of Nicholson’s blowflies as an example, Ecology, 73, 1552-1563, (1992)
[5]	Beretta, E.; Kuang, K., Global analyses in some delayed ratio-dependent predator – prey systems, Nonlinear anal., 32, 381-408, (1998) · Zbl 0946.34061
[6]	Arditi, R.; Ginzburg, L.R., Coupling in predator – prey dynamics: ratio-dependence, J. theor. biol., 139, 311-326, (1989)
[7]	Hanski, I., The functional response of predator: worries about scale, Tree, 6, 141-142, (1991)
[8]	Cushing, J.M., Integro-differential equations and delay models in population dynamics, () · Zbl 0363.92014
[9]	Kuang, K., Delay differential equations with applications in population dynamics, (1993), Academic Press New York · Zbl 0777.34002
[10]	Freedman, H.I.; Rao, V.S.H., The tradeoff between mutual interference and time lag in predator – prey models, Bull. math. biol., 45, 991-1004, (1983) · Zbl 0535.92024
[11]	Freedman, H.I.; So, J.; Waltman, P., Coexistence in a model of competition in the chemostat incorporating discrete time delays, SIAM J. appl. math., 49, 859-870, (1989) · Zbl 0676.92013
[12]	Wang, W.; Ma, Z., Harmless delays for uniform persistence, J. math. anal. appl., 158, 256-268, (1991) · Zbl 0731.34085
[13]	Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Dordrecht · Zbl 0752.34039