On an eco-epidemiological model with prey harvesting and predator switching: Local and global perspectives

https://doi.org/10.1016/j.nonrwa.2010.02.012Get rights and content

Abstract

In this paper, we study an eco-epidemiological model where prey disease is modeled by a Susceptible-Infected (SI) scheme. Saturation incidence kinetics is used to model the contact process. The predator population adapt switching technique among susceptible and infected prey. The prey species is supposed to be commercially viable and undergo constant non-selective harvesting. We study the stability aspects of the basic and the switching models around the infection-free state and the infected steady state from a local as well as a global perspective. Our aim is to study the role of harvesting and switching on the dynamics of disease propagation and/or eradication. A comparison of the local and global dynamical behavior in terms of important system parameters is obtained. Numerical simulations are done to illustrate the analytical results.

Introduction

Theoretical research and field observations have established the prevalence of various infectious diseases amongst the majority of ecosystem populations. This has necessitated the study of the impact of epidemiological parameters in the ecological domain. Eco-epidemiology is the branch of bio-mathematics that merges the above two otherwise independent fields to understand the dynamics of disease propagation on the prey–predator population. Modeling studies on such eco-epidemiological problems have grown enormously in the recent past [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. These studies have emphasized the role of disease in regulating various ecological aspects of the populations concerned.

Modeling studies on disease dominated ecological populations have addressed issues like disease related mortality, reduction in reproduction, change in population sizes and disease induced oscillation of population states [13], [8] Anderson and May [1] formulated a prey–predator model with prey infection and observed destabilization due to infection. Hadeler and Freedman [2] studied an SI modification of the Rosenzweig prey–predator model and obtained a threshold above which an infected equilibrium or an (infected) periodic solution exists. Venturino analyzed prey–predator models with disease in the prey [3] and the predator [7]. The role of prey infection on the stability aspects of a prey–predator model with different functional responses was studied by Bairagi et al. [12]. Haque and Chattopadhyay [14] investigated the role of transmissible diseases in a prey-dependent prey–predator system with prey infection. Han et al. [15] analyzed four eco-epidemiological models for SIS and SIR diseases with standard and mass action incidents. The existence of strange attractors in an eco-epidemiological model with standard incidence was demonstrated by Hilker and Malchow [16].

Contemporary modeling literature on eco-epidemiology has already established that predators have a natural tendency to consume a disproportionate number of infected prey [17]. In most cases, parasite infection causes a change in the behavioral pattern of the prey that make them more vulnerable to predation. Due to infection, the prey species tends to live in locations that are easily accessible to the predator. There are many examples of infected fish or aquatic snails living close to the water surface and snails living on the top of vegetation. Moreover, the infected prey may become weaker, so they are easily caught by the predator [18]. Peterson and Page [19] observed that wolf attacks on moose in Isle Royale in Lake Superior are more successful when the moose are infected with lungworm. Predators like lions or wolves have a natural tendency to select prey that are weakened by a disease [20], [21]. Lafferty and Morris [22] observed experimentally that predation rates of piscivorous birds on infected fish is much higher than that on susceptible fish.

The above-mentioned tendency of preferential predation would cause a substantial reduction in the number of infected prey which in turn would compel the predator to shift its attention towards susceptible prey temporarily [23]. This predatorial characteristic of changing the prey type exemplifies the well-known phenomenon of predator switching. A number of modeling studies have been performed that involve this switching phenomenon [24], [25], [26], [27], [28], [29]. Malchow et al. [30] investigated an excitable plankton ecosystem model with virus infection that includes predator switching using a Holling type-III functional response. Bhattacharyya and Mukhopadhyay [29] analyzed the effect of predator switching with and without prey group defence under non-homogeneous mixing of the population. They also performed a mathematical analysis of the bifurcating solutions of a prey–predator model with switching predator [31].

Commercial exploitation of ecological resources to meet the growing needs of society has been a topic of much concern for ecologists, bio-economists and natural resource managers. The study of population dynamics with harvesting is a topic of mathematical bio-economics and is mainly concerned with optimal management of renewable resources [32], [33]. Harvesting is commonly practised in fisheries, forestry and in wildlife management. It has a significant effect on the dynamical evolution of the harvested species; it can have a stabilizing effect, a destabilizing effect and even an oscillation inducing effect [34], [35], [36]. The effect of constant rate harvesting has been investigated by many authors [37], [38], [39], [40], [41], [42], [43], [44], [45], [46]. These investigations revealed very rich and interesting dynamics such as stability of equilibria, existence of Hopf bifurcation, limit cycles, homoclinic loops, Bogdanov–Takens bifurcation and even catastrophe. Population exploitation in the presence of parasites can have an even more significant impact on the population dynamics. Parasites can reduce the abundance as well as yield by increasing the mortality, reducing the fecundity and creating a decline in marketability of the harvested stock [47], [48]. Moreover, parasites can be eliminated locally from a population by reducing the hosts’ density [49]. Bairagi et al. [50] studied an eco-epidemiological model with harvesting on the susceptible and infected prey. They have analyzed the role of harvesting in regulating the cyclic behavior of the system, and in eliminating prey infection, together with its overall impact on the system dynamics.

In the present research, we formulate a mathematical model of prey–predator interaction where the prey suffers micro-parasite infection. It is conjectured that due to the ecologically well-established phenomenon of preferential predation on infected prey, a substantial reduction in their number could occur which, in turn, would compel the predator to switch to susceptible prey. It is also assumed that the prey species has a commercial value and both the susceptible and infected prey undergo harvesting at a constant rate. We first study the stability behavior of the basic model (without switching) in order to obtain parameter thresholds that can cause (i) eradication of prey infection and (ii) disease persistence. Then we study the model with predator switching from the same perspective in an attempt to compare the role of harvesting in the presence and absence of the switching behavior of the predator.

Section snippets

Model formulation

The basic model consists of two population subclasses—(i) prey population with density N(t) and (ii) predators having population concentration Y(t). We make the following assumptions.

  • The prey population grows logistically with intrinsic growth rate r and environmental carrying capacity k.

  • Due to micro-parasite infection, the prey population is assumed to be divided into two subclasses, namely the susceptible prey (S(t)) and the infected prey (I(t)); that is, at any instant of time t, N(t)=S(t)+I

Stability study

System (2.1) possesses the following equilibria.

  • 1.

    The trivial equilibrium ET=(0,0,0).

  • 2.

    The axial equilibrium EA={k(1h1r),0,0}.

  • 3.

    The boundary equilibrium EB=(S1,0,Y1), where S1=md2qp1d2Y1=mqqp1d2{r(1md2k(qp1d2))h1}.

  • 4.

    The equilibrium point of coexistence E=(S,I,Y), where (S,I,Y) is the positive solution (if one exists) of the coupled system r(1x+yk)βy1+αyp1zm+xh1=0βx1+αyd1p2zm+yh2=0d2+q(p1xm+x+p2ym+y)=0.

The axial equilibrium point exists if r>h1. Stability analysis around EA shows

The model with switching predator

In this section, we investigate a modification of the basic model that incorporates switching of the predator among susceptible and infected prey. dSdt=rS(1S+Ik)βSI1+αIp1SY1+(IS)2h1SdIdt=βSI1+αId1Ip2IY1+(SI)2h2IdYdt=d2Y+qp1SY1+(IS)2+qp2IY1+(SI)2. The functions [p1SY1+(I/S)2] and [p2IY1+(S/I)2] mathematically characterize the switching mechanism. From an ecological viewpoint these functions signify that the predation rate on a species decreases when the population density of that species

Concluding remarks and numerical results

In this paper, we have studied a mathematical prey–predator model at the interface of ecology, epidemiology and bio-economics. Switching is a well-established predatorial characteristic in the ecological domain; micro-parasite infection among an ecological population is a very common epidemiological feature; and excessive species exploitation is a topic of serious present-day concern. We have combined these three aspects in our modeling study. The main objective of the work is to investigate

Acknowledgement

The authors are grateful to the reviewers for their valuable suggestions, which have improved the scientific content and presentation of the work. The authors also acknowledge financial support from the Department of Science and Technology, Ministry of Human Resource Development, Government of India (Grant No. SR/S4/MS:296/05).

References (51)

  • H. Malchow et al.

    Spatiotemporal patterns in an excitable plankton system with lysogenic viral infections

    Math. Comput. Modelling

    (2005)
  • B. Mukhopadhyay et al.

    Bifurcation analysis of an ecological food-chain model with switching predator

    Appl. Math. Comput.

    (2008)
  • K.S. Chaudhuri

    A bio-economic model of harvesting a multispecies fishery

    Ecol. Model.

    (1986)
  • B.S. Goh et al.

    Optimal control of a prey–predator system

    Math. Biosci.

    (1974)
  • M. Mesterton-Gibbons

    A technique for finding optimal two-species harvesting policies

    Nat. Resour. Model.

    (1996)
  • A.P. Dobson et al.

    The effects of parasites on fish populations—Theoretical aspects

    Int. J. Parasitol.

    (1987)
  • N. Bairagi et al.

    Harvesting as a disease control measure in an eco-epidemiological system—A theoretical study

    Math. Biosci.

    (2009)
  • R.M. Anderson et al.

    The invasion, persistence and spread of infectious diseases within animal and plant communities

    Philos. Trans. R. Soc. Lond.

    (1986)
  • K.P. Hadeler et al.

    Predator–prey populations with parasite infection

    J. Math. Biol.

    (1989)
  • E. Venturino

    Epidemics in predator–prey models: Diseases in the prey

  • E. Venturino

    Epidemics in predator–prey models: Disease in the predator

    IMA J. Math. Appl. Med. Biol.

    (2002)
  • S.R. Hall et al.

    Selective predation and productivity jointly drive complex behavior in host-parasite systems

    Amer. Nat.

    (2005)
  • B. Mukhopadhyay et al.

    Dynamics of a delayed epidemiological model with nonlinear incidence: The role of infected incidence fraction

    J. Biol. Syst.

    (2005)
  • A. Fenton et al.

    The impact of parasite manipulation and predator foraging behavior on predator–prey communities

    Ecology

    (2006)
  • J. Mena-Lorca et al.

    Dynamic models of infectious diseases as regulators of population sizes

    J. Math. Biol.

    (1992)
  • Cited by (0)

    View full text