Mathematical Medicine and Biology Advance Access originally published online on August 12, 2008
Mathematical Medicine and Biology 2008 25(3):247-266; doi:10.1093/imammb/dqn017
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Predator–prey model with disease infection in both populations
Department of Public Health and Biostatistics Center, China Medical University, 91 Hsueh-Shih Road, Taichung, Taiwan 404
Department of Applied Mathematics, National Chung Hsing University, Taichung, Taiwan
Email: hsieh{at}mail.cmu.edu.tw
Received on October 15, 2007. Revised on June 22, 2008. Accepted on July 2, 2008.
A predator–prey model with disease infection in both populations is proposed to account for the possibility of a contagious disease crossing species barrier from prey to predator. We obtain several threshold parameters from local analysis of various equilibria of the proposed system as well as coupled conditions on these threshold parameters which determine the stability of these equilibria. One of the coupled conditions, in the form of an ecological threshold number for the predator–prey ecosystem, always determines the coexistence of predators and prey. The other condition, in the form of a disease basic reproduction number, dictates whether the disease will become endemic in the ecosystem. Under one combination of these coupled conditions, a highly infectious disease could drive the predators to extinction when predators and prey would have coexisted without the disease. For another combination of the conditions, the predation of the more vulnerable infected prey could cause the disease to be eradicated in the ecosystem, in some case even approaching a disease-free periodic solution, when the disease would have otherwise remained endemic in the prey population in the absence of predation. This indicates that the presence of disease in both predators and prey could either promote or impair coexistence, and its precise impact needs to be explored specifically in each particular situation. By considering disease infection in both populations, our model also yields more complex dynamics, allowing for the possibility of bistability and periodic oscillation, in either disease-free or endemic states, in the ecosystem for which the conditions are obtained analytically and with the help of numerical simulations.
Keywords: species barrier; predator–prey coexistence; ecological threshold parameter; disease basic reproduction number; positive equilibrium; periodic oscillation