×

zbMATH — the first resource for mathematics

A model of predator-prey dynamics as modified by the action of a parasite. (English) Zbl 0698.92024
Summary: A predator-prey population is described in which the prey population may be either a secondary host or a primary host to a parasite, but the predator is always a primary host. Those prey that have been invaded by the parasite have their behavior modified so as to make them more susceptible to predation. The model is described by a system of three autonomous ordinary differential equations. Conditions for persistence of all populations are given in the case that both populations are primary hosts. A brief discussion of the stability of the interior equilibrium is given.

MSC:
92D25 Population dynamics (general)
37N99 Applications of dynamical systems
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D99 Stability theory for ordinary differential equations
37C75 Stability theory for smooth dynamical systems
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, R.M.; May, R.M., Regulation and stability of host-parasite population interactions, J. animal ecol., 47, 219-247, (1978)
[2] Anderson, R.M.; May, R.M., The invasion and spread of infectious diseases within animal and plant communities, Phil. trans. R. soc. lond., B314, 533-570, (1986)
[3] Arme, C.; Owen, R.W., Infections of the three-spined stickleback, gasterosteus aculeatus L., with the plerocercoid larvae of schistocephalus solidus (muller 1776), Parasitology, 57, 301-314, (1967), with special reference to pathological effects
[4] Butler, G.J.; Freedman, H.I.; Waltman, P., Uniformly persistent systems, Proc. am. math. soc., 96, 425-430, (1986) · Zbl 0603.34043
[5] Dobson, A.P., The population biology of parasite-induced changes in host behavior, Quart. rev. biol., 63, 139-165, (1988)
[6] Freedman, H.I., Deterministic mathematical models in population ecology, (1987), HIFR Consulting Ltd., Edmonton · Zbl 0448.92023
[7] Freedman, H.I.; Addicott, J.F.; Rai, B., Obligate mutualism with a predator: stability and persistence of three-species models, Theor. pop. biol., 32, 157-175, (1987) · Zbl 0638.92020
[8] Freedman, H.I.; So, J.W.-H., Global stability and persistence of simple food chains, Math. biosci., 76, 69-86, (1985) · Zbl 0572.92025
[9] Freedman, H.I.; Waltman, P., Predator influence on the growth of a population with three genotypes, J. math. biol., 6, 367-374, (1978) · Zbl 0393.92012
[10] Freedman, H.I.; Waltman, P., Persistence in models of three interacting predator-prey populations, Math. biosci., 68, 213-231, (1984) · Zbl 0534.92026
[11] Graham, G.L., The behaviour of beetles, tribolium confusum, parasitized by the larvae stage of a chicken tapeworm, raillietina cesticullus, Trans. am. microsc. soc., 85, 163, (1966)
[12] K. P. Hadeler and H. I. Freedman, Predator-prey populations with parasitic infection,J. Math. Biol. · Zbl 0716.92021
[13] Holmes, J.C.; Bethel, W.M., Modification of intermediate host behaviour by parasites, (), 123-149, Suppl. No. 1 to the Zool. J. Linnean Soc.
[14] Mueller, J.F., Host-parasite relationships as illustrated by the cestode, spirometra mansonoides, (), 15-58, Proc. 26th Annual Biology Colloquium
[15] Peterson, R.O., Wolf ecology and prey relationships on isle royale, Natl. park serv. sci. monogr., 11, (1977)
[16] Rau, M.E.; Caron, F.R., Parasite-induced susceptibility of moose to hunting, Can. J. zool., 57, 2466-2468, (1979)
[17] Tiner, J.D., The migration, distribution in the brain, and growth of ascarid larvae in rodents, J. infect. dis., 92, 105-113, (1953)
[18] Tiner, J.D., The fraction of peromyscus leucopus fatalities caused by raccoon ascarid larvae, J. mammal., 35, 589-592, (1954)
[19] Waltman, P., Deterministic: threshold models in the theory of epidemics, (), No. 1 · Zbl 0293.92015
[20] Williams, H.H., Helminth diseases of fish, Helminth abstr., 36, 261-295, (1967)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.