Modeling and analysis of a predator-prey model with disease in the prey. (English) Zbl 0978.92031

Summary: A system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. Mathematical analyses of the model equations with regard to invariance of non-negativity, boundedness of solutions, nature of equilibria, permanence and global stability are analyzed. If the coefficient in conversing prey into predator \(k=k_0\) is constant (independent of delay \(\overline\tau\), gestation period), we show that positive equilibrium is locally asymptotically stable when time delay \(\overline\tau\) is suitably small, while a loss of stability by a Hopf bifurcation can occur as the delay increases. If \(k=k_0 e^{-d\overline \tau}\) \((d\) is the death rate of predator), numerical simulation suggests that time delay has both destabilizing and stabilizing effects, that is, positive equilibrium, if it exists, will become stable again for large time delays. A concluding discussion is then presented.


92D40 Ecology
92D30 Epidemiology
34K20 Stability theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
Full Text: DOI


[1] Kermack, W. O.; McKendrick, A. G., Contributions to the mathematical theory of epidemics, Proc. Roy. Soc. A, 115, 700 (1927) · Zbl 0007.31502
[3] Anderso, R. M.; May, R. M., Infectious Disease of Humans, Dynamics and Control (1991), Oxford University: Oxford University Oxford
[4] Bailey, N. J.T., The Mathematical Theory of Infectious Disease and its Applications (1975), Griffin: Griffin London
[5] Diekmann, O.; Hecsterbeck, J. A.P.; Metz, J. A.J., The legacy of Kermack and McKendrick, (Mollision, D., Epidemic Models, their Structure and Relation to Data (1994), Cambridge University: Cambridge University Cambridge) · Zbl 0839.92018
[6] Hadeler, K. P.; Freedman, H. I., Predator-prey population with parasitic infection, J. Math. Biol., 27, 609 (1989) · Zbl 0716.92021
[7] Chattopadhyay, J.; Arino, O., A predator-prey model with disease in the prey, Nonlinear Anal., 36, 749 (1999) · Zbl 0922.34036
[8] Venturino, E., The influence of disease on Lotka-Volterra systems, Rockymount. J. Math., 24, 389 (1994)
[10] Peterson, R. O.; Page, R. E., Wolf density as a predictor of predation rate, Swedish Wildlife Research Suppl., 1, 771 (1987)
[11] Wang, W. D.; Chen, L. S., A predator-prey system with stage-structure for predator, Comp. Math. Appl., 33, 45, 83 (1997)
[12] Zhao, T.; Kuang, Y.; Smith, H. L., Global existence of periodic solutions in a class of delayed gause-type predator-prey systems, Nonlinear Anal., 28, 1373 (1997) · Zbl 0872.34047
[13] Anderson, R. M.; May, R. M., The population dynamics of microparasites and their invertebrates hosts, Proc. R. Soc. Lond. B, 291, 451 (1981)
[14] Ma, W. B.; Takeuchi, Y., Stability analysis on a predator-prey system with distributed delays, J. Comput. Anal. Math., 88, 79 (1998) · Zbl 0897.34062
[15] Beretta, E.; Kuang, Y., Global analyses in some delayed ratio-dependent predator-prey systems, Nonlinear Anal., 32, 3, 381 (1998) · Zbl 0946.34061
[16] Yang, X.; Chen, L. S.; Chen, J. F., Permanence and positive periodic solution for the single-species nonautonomous delay diffusive model, Comput. Math. Appl., 32, 109 (1996) · Zbl 0873.34061
[17] Kuang, Y., Delay Differential Equation with Application in Population Dynamics (1993), Academic Press: Academic Press New York
[18] Freedman, H. I.; Sree Hari Rao, V., The trade-off between mutual interference and time lags in predator-prey-systems, Bull. Math. Biol., 45, 991 (1983) · Zbl 0535.92024
[19] Hale, J. K., Theory of Functional Differential Equations (1977), Springer: Springer NewYork · Zbl 0425.34048
[20] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. N., Theory and Application of Hopf Bifurcation (1981), Cambridge University: Cambridge University Cambridge · Zbl 0474.34002
[21] Hale, J. K.; Waltman, P., Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20, 388 (1989) · Zbl 0692.34053
[22] Brauer, F.; Ma, Z., Stability of stage-structured population models, J. Math. Anal. Appl., 126, 2, 301 (1987) · Zbl 0634.92014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.