Qualitative Analysis for a Reaction-Diffusion Predator-Prey Model with Disease in the Prey Species

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 236208, 9 pages http://dx.doi.org/10.1155/2014/236208 Research Article Qualitative Analysis for a Reaction-Diffusion Predator-Prey Model with Disease in the Prey Species Meihong Qiao,1 Anping Liu,1 and Urszula ForyV2 1 2 School of Mathematics & Physics, China University of Geoscience, Wuhan 430074, Hubei Province, China Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics, & Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland Correspondence should be addressed to Meihong Qiao; Received 3 February 2014; Accepted 8 April 2014; Published 29 April 2014 Academic Editor: Zhijun Liu Copyright © 2014 Meihong Qiao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A diffusive predator-prey system with disease in predator species and no-flux boundary condition is considered. Sufficient conditions which ensure persistence of the system are obtained. Conditions of disease-free ecosystem are also studied. Furthermore, sufficient conditions for global asymptotic stability of the unique positive equilibrium and disease-free equilibrium of the system are derived using the approach of Lyapunov function. 1. Introduction Ecoepidemiology is a relatively new branch of study in theoretical biology, which tackles problems by dealing with both ecological and epidemiological approach. It can be viewed as the coupling of an ecological predator-prey or competition model with an epidemiological SI, SIS, or more complex model. Anderson and May [1] were the first who marked that the effect of disease in ecological systems is an important issue from both mathematical and ecological point of view. They proposed an ecoepidemiological model by merging the ecological predator-prey model introduced by Lotka and Volterra with epidemiological model. Clearly, in the natural world, species does not exist alone. While the disease is spread within the species, the species also competes with other species for environmental resources like space or food, or is predated by other species. Therefore, it is of more biological significance to consider the effect of interacting species when we study the dynamical behaviors of epidemiological models. Many papers have been devoted to study the effects of a disease on a predator-prey system. Venturino [2] studied SI and SIS models with disease spread among the prey when the logistic growth of both the prey and predator populations is assumed and the predators eat infected preys only. In [3], Hsu and Huang considered the following predator-prey model: 𝑢 𝑑𝑢 = 𝑟𝑢 (1 − ) − 𝑝 (𝑢) V, 𝑑𝑡 𝐾 𝑑V ℎV = 𝑠V (1 − ) , 𝑑𝑡 𝑢 (1) where 𝑢 and V represent densities of the populations of prey and predators, respectively, and 𝑟, 𝑠, 𝐾, and ℎ are positive constants. The population of prey grows logistically with carrying capacity 𝐾 and intrinsic growth rate 𝑟 in the absence of predation. Predators consume prey according to the functional response 𝑝(𝑢) and grow logistically with intrinsic growth rate 𝑠. Carrying capacity of the predator species is proportional to the size of the prey population. It should be noticed that the model described by (1) is a generalisation of the prey-predator model proposed by May [4] which is known as Holling-Tanner model. In this model the functional response 𝑝(𝑢) = 𝑎𝑢/(𝑏 + 𝑢) is of the Holling type [5], and it is one of the prototype models involving limit cycle dynamics. 2 Journal of Applied Mathematics 2. Model Formulation On the basis of (1) we propose an ecoepidemiological model with a disease spread in the predator population. We assume that only predator can be infected and the infected individual does not recover or become immune. Because the predation ability of healthy (and susceptible at the same time) predators is stronger than infected ones, we suppose that prey can be preyed on only by healthy predators. Moreover, we also assume the simplest linear form of functional response 𝑝(𝑢) = 𝑘𝑢. Therefore, the model reads 𝑢 𝑑𝑢 = 𝑟𝑢 (1 − ) − 𝑘𝑢V, 𝑑𝑡 𝐾 (2) 𝑑𝑤 = 𝛽𝑤V − 𝑑𝑤 − 𝜇𝑤2 , 𝑑𝑡 where 𝑢, V, and 𝑤 represent densities of the populations of prey, susceptible predator, and infected predator, respectively. The death rate of infected predators equals 𝑑, 𝛽 is the infectious rate of the disease, and 𝜇 is the density-dependent death rate of infected predators. Other parameters are the same as in (1). Species dispersal is one of the most prevalent phenomena of nature, and many empirical studies and monographs on population dynamics in a spatial heterogeneous environment have been done (see [6–15] and the references cited therein). Most important subjects of population diffusion models are coexistence of populations, local and global stability of equilibria, existence of periodic solutions, and so forth, (see [16– 20]). In particular, single population models were considered, for example, in [21–23], while predator-prey system with the prey dispersal was studied, for example, in [24–26]. Such type of model is still of great interest and importance; compare the recent papers [27–30] and the references therein. Taking into account inhomogeneous distribution of predators and their prey in different spatial locations within a fixed bounded domain Ω in R𝑁 with smooth boundary at any given time and the natural tendency of each species to diffuse to areas of smaller population density, we are led to consider the following reaction-diffusion system: 𝜕V ℎV = 𝑑2 ΔV + 𝑠V (1 − ) − 𝛽𝑤V, 𝜕𝑡 𝑢 𝜕𝑤 = 𝑑3 Δ𝑤 + 𝛽𝑤V − 𝑑𝑤 − 𝜇𝑤2 , 𝜕𝑡 𝜕𝑢 𝜕V 𝜕𝑤 = = = 0, 𝜕𝜂 𝜕𝜂 𝜕𝜂 𝑢 (𝑥, 0) = 𝑢0 (𝑥) ≥ 0, 𝑤 (𝑥, 0) = 𝑤0 (𝑥) ≥ 0, 𝑥 ∈ Ω, 𝑠 󳨃󳨀→ 𝑠, 𝑟 𝑥 ∈ Ω, V (𝑥, 0) = V0 (𝑥) ≥ 0, 𝑑1 󳨃󳨀→ 𝑑1 , 𝑟 𝑠ℎ 󳨃󳨀→ 𝑏, 𝑟𝐾 𝛽 󳨃󳨀→ 𝛽, 𝑟 𝑑3 󳨃󳨀→ 𝑑3 , 𝑟 (5) 𝜇 𝑑 󳨃󳨀→ 𝑑, 󳨃󳨀→ 𝜇, 𝑟 𝑟 obtaining nondimensional version of the model 𝜕𝑢 = 𝑑1 Δ𝑢 + 𝑢 (1 − 𝑢) − 𝑘𝑢V, 𝜕𝑡 𝑥 ∈ Ω, 𝜕V V = 𝑑2 ΔV + V (𝑠 − ) − 𝛽𝑤V, 𝜕𝑡 𝑢 𝑥 ∈ Ω, 𝜕𝑤 = 𝑑3 Δ𝑤 + 𝛽𝑤V − 𝑑𝑤 − 𝑤2 , 𝜕𝑡 𝑥 ∈ Ω, 𝜕𝑢 𝜕V 𝜕𝑤 = = = 0, 𝜕𝜂 𝜕𝜂 𝜕𝜂 𝑢 (𝑥, 0) = 𝑢0 (𝑥) ≥ 0, 𝑤 (𝑥, 0) = 𝑤0 (𝑥) ≥ 0, (6) 𝑥 ∈ 𝜕Ω, V (𝑥, 0) = V0 (𝑥) ≥ 0, 𝑥 ∈ Ω. In this paper we assume that 𝑁 = 1 as calculations are simpler in such a case. However, the results presented below can be extended for 𝑁 > 1. Our goal is to give conditions guaranteeing persistence of the ecosystem described by (6). The persistence means that the disease is spread and endemic equilibrium appears. Equations (6) have positive (endemic) equilibrium (𝑢∗ , V∗ , 𝑤∗ ), where 𝑢∗ = 1 (𝛽2 − 𝑘𝑑𝛽 − 𝑘𝑠 − 1 2𝛽2 2 +√(𝛽2 − 𝑘𝑑𝛽 − 𝑘𝑠 − 1) + 4𝛽2 ) , (7) 1 − 𝑢∗ , 𝑤∗ = 𝛽V∗ − 𝑑. 𝑘 Notice that V∗ > 0 ⇔ 𝑢∗ < 1 and 𝑤∗ > 0 ⇔ V∗ > 𝑑/𝛽 ⇔ 𝑢∗ < 1 − 𝑑𝑘/𝛽, which means that 𝛽 > 𝑑𝑘 is the necessary condition for the existence of the positive equilibrium. One can easily check that 𝑢∗ < 1, while the inequality ∗ V > 𝑑/𝛽 is equivalent to 𝛽 > 𝑑𝑘 + 𝑑/𝑠. On the other hand, it is a (...truncated)