The Holling-Tanner model for predator-prey systems is adapted to incorporate
the spread of disease in the prey. The analysis of the dynamics centers on
bifurcation diagrams in which the disease transmission rate is the primary
parameter. The ecologically reasonable assumption that the diseased prey
are easier to catch enables tractable analytic results to be obtained for
the stability of the steady states and the locations of Hopf bifurcation
points as a function of the ecological parameters. Two parameters of
particular relevance are the ratio of the predator's intrinsic growth rate
to the prey's growth rate and the maximum number of infected prey that can
be eaten per time. The dynamics are shown to be qualitatively different
depending on the comparative size of these parameters. Numerical results
obtained with AUTO are used to extend the local analysis and further
illustrate the rich dynamics.
Citation: Peter A. Braza. Predator-Prey Dynamics with Disease in the Prey[J]. Mathematical Biosciences and Engineering, 2005, 2(4): 703-717. doi: 10.3934/mbe.2005.2.703
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Abstract
The Holling-Tanner model for predator-prey systems is adapted to incorporate
the spread of disease in the prey. The analysis of the dynamics centers on
bifurcation diagrams in which the disease transmission rate is the primary
parameter. The ecologically reasonable assumption that the diseased prey
are easier to catch enables tractable analytic results to be obtained for
the stability of the steady states and the locations of Hopf bifurcation
points as a function of the ecological parameters. Two parameters of
particular relevance are the ratio of the predator's intrinsic growth rate
to the prey's growth rate and the maximum number of infected prey that can
be eaten per time. The dynamics are shown to be qualitatively different
depending on the comparative size of these parameters. Numerical results
obtained with AUTO are used to extend the local analysis and further
illustrate the rich dynamics.