A predator–prey model with disease in the predator species only
Introduction
Mathematical and statistical models can help us to identify key parameters which determine the rich dynamics of an ecological or epidemiological system. In the development of quantitative theory for interaction of predator and prey, mathematical ecology is also an important factor along with the experimental ecology, see [1] and references therein. Transmissible disease in an ecological situation is fast becoming a major field of study in its own right. The first mathematical description of contagious diseases has been formulated by Kermack and McKendric [2]. Both theoretical and experimental investigations in these two fields namely ecology and epidemiology progressed independently along the years, until the late eighties and early nineties. The ecological literature has increasingly emphasized the importance of parasites in shaping the dynamics of both plant and animal communities, [3], [4], [5].
Some models merging features of the two phenomena, i.e. the demographics of interacting species and an epidemic evolution in such a composite environment, were then considered, see [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17] and the references therein.
Most of the above models are based on the assumption that the infected prey is more vulnerable to predation except for those developed by Venturino [11], and Haque and Venturino [15]. In these two papers the authors considered the case when disease is spreading among the predator population. The model considered here also studies the situation where an epidemic runs among the predator population but differs from previous models; it includes the recovery rate and also takes into account also the stability of the positive interior equilibrium point. In addition, we assume that the predator has a logistic growth rate since it has sufficient resources for alternative foods; and it is argued that alternative food sources may have an important role in promoting the persistence of predator–prey systems. van Baalen et al. [18] have observed that the switching of the predator species has a significant contribution to the persistence of species in the predator–prey system. They also have analyzed the conditions for stability as well as long-term behavior of the system under bounded population fluctuations. This leads to specific hypotheses about which types of alternative food (in terms of nutritional quality, availability, and handling time) promote persistence and by which mechanism.
Section snippets
Predator–prey model with disease in the predator species only
Let us consider a predator–prey model, where represents the number of prey and denotes the number of predators at the time . The classical well known model is given by Here is the growth rate of the prey and is the prey carrying capacity of the environment. The total predation rate is and the conversion rate is denoted by . Again represents the growth rate of the predator and is the predator carrying capacity of the environment.
Now
Stability analysis
Here we perform the stability and bifurcation analysis of the system (2.4). System (2.4) can be written in the form where , and the Jacobian matrix of the system is where , and .
Behavior of the proportional system (2.4) and the absolute system (2.3)
The behavior of the system when one or more of the susceptible predator, infected predator and prey populations are initially absent has already been discussed in Section 3. For starting values for which all three types of individuals are initially present note that for all parameter values there is exactly one LAS equilibrium. The following remarks can be done.
Remark 2 For the proportional system (2.4) with and we have the following results: (i) For and the system
The role of infection on the predator–prey system
To observe the influence of infection in the predator–prey system we set in (2.4), and obtain a simple Lotka–Volterra type predator–prey system, with density-dependent logistic growth for the prey population, where the predator has an alternative food source. However it is analyzed by Bazykin et al. [22], but here we report the results in terms of our system parameters for comparison purpose. This system has four biologically relevant equilibria namely : (i) , (ii) , (iii)
The effect of prey removal on the eco-epidemiological system
To observe the role of the predator in the eco-epidemiological system we put in our system (2.4) and get a reduced system which is well known and has been analyzed extensively by Diekmann and Heesterbeek [23], Hethcote [24] and Brauer and Castillo-Chavez [25]. Here the local stability results have been reported in terms of our system parameters for comparison purposes. This system has four equilibria, namely (i) which is always admissible, (ii) , which is always
Numerical investigations
Many biological examples are given in the Table 2 of [19], here we take the example of rabbit/fox predator–prey system as in the UK, the European rabbit plays a vital role in maintaining bio-diversity and the fox (Vulpus vulpus) has become the main carrier of rabies in Europe. Now let the fox be infected by rabies. The disease is transmitted mostly as a result of an infected animal biting a healthy animal. The birth rate of foxes , its natural death rate , transmission coefficient of rabies
Discussion
In this article we have proposed and analyzed a mathematical model that consists of three non-linear autonomous differential equations for three different populations namely prey (), susceptible predator (), infected predator (). The conditions for existence and stability of the equilibria of the system have been given. The bifurcation situations have also been observed around important equilibrium points.
We have investigated that there are five epidemiological threshold quantities for our
Acknowledgments
I am very grateful to the referees for their comments. I am greatly indebted to John R. King (editor of IMA Journal of Mathematics in Medicine and Biology, associate editor of SIAM Journal of Applied Mathematics), from whom I have learned applications of mathematics to biology and medicine. Finally, I am also thankful to Professor Philip Maini, Mathematical Institute(CMB), University of Oxford, who saw the earlier version of the manuscript. Finally, thanks go to the UK-IERI for financial
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