Intra-specific competition in predator can promote the coexistence of an eco-epidemiological model with strong Allee effects in prey

Highlights

• We study dynamics of eco-epidemiological system with disease and Allee effect in prey and Hydra effect in predator.

• Disease transmission follows the standard incidence.

• Perform local and global dynamics of the model equations.

• Competition in the predator can promote the coexistence of all the three populations.

Abstract

An eco-epidemiological model with Allee effects and disease in prey has been proposed and analyzed. The proposed model incorporates intra-specific competition in predator due to the limited food source, and assumes standard incidence disease transmission. We analyzed the corresponding submodels with and without the Allee effects to obtain the complete dynamics of the full model. Our results show that our full model shows multi-stability between the planner equilibriums (where the susceptible prey co-exists with infected prey or predator); both the full model and its submodels exhibit the hydra effects caused by the intra-specific competition in predator. We determined the existence of multiple interior attractors and their stability. Our analysis shows that our system has at most two interior equilibria whose stability is either both saddle or one stable with another one saddle. One of the most interesting findings is that the competition in the predator can promote the coexistence of all the three populations. In addition, we discussed how the frequency-dependent transmission differs from the model with the density-dependent transmission and compare the hydra effects observed in our model to other existing models in literature.

Introduction

An Allee effect is a natural phenomenon describing a positive correlation between the population size/density and the per-capita growth rate (pgr) at low population densities (Allee, 1931, Odum and Allee, 1954). Component Allee effects are measurable ecological components that increase with population size. The synergy of component Allee effects and negative density factors such as competition can result in demographic Allee effects (Stephens et al., 1999, Courchamp et al., 2008). When a species experiences a demographic Allee effect, there is a critical threshold population density (known as Allee threshold) below which the pgr becomes negative and extinction becomes an almost certain event; above which the pgr is positive and the species may sustain. Due to the significant biological relevance of Allee effects, the concept of Allee effects receives substantial attention from both theoretical and applied ecologists. Allee effects have great impacts in species’ establishment, persistence, invasion (Amarasekare, 1998, Wang et al., 2002, Wang et al., 2011b, Drake, 2004, Taylor and Hastings, 2005, Shi and Shivaji, 2006, Berezovskaya et al., 2010) and evolutionary traits (Cushing and Hudson, 2012). Empirical evidence of the Allee effect has been reported in many natural populations, including plants (Groom, 1998, Ferdy et al., 1999), insects (Kuussaari et al., 1998), marine invertebrates (Stoner and Ray-Culp, 2000), birds and mammals (Courchamp et al., 2000a). For details of the Allee effects we refer the reader to see the reviews of Courchamp et al. (2008), William (2010) and the references therein. Allee effects on population interaction have been studied by many researchers [e.g., see (Schreiber, 2003, Zhou et al., 2004, Jang, 2011, Kang and Yakubu, 2011, Wang et al., 2011a, Kang et al., 2014b). Disease is known as one of the basic reasons for species extinction. When disease is coupled with Allee effects then the systems are more prone to extinction (Hilker et al., 2007). The combined impact of disease and the Allee effect are observed in African wild dog Lycaon pictus (Courchamp et al., 2000b) and island fox Urocyon littoralis (Angulo et al., 2007). Recently much research has been done on Allee effects in the presence of disease (Yakubu, 2007, Hilker et al., 2009, Thieme et al., 2009, Sasmal and Chattopadhyay, 2013, Kang et al., 2014a, Sasmal et al., 2014). These studies suggest that Allee effects have important roles in population dynamics, especially when it couples with disease.

It has long been known that increasing a predator's mortality rate can increase its population size (Rosenzweig and MacArthur, 1963). A review by Abrams (2009) discusses how the greater mortality rate increases the population size. This phenomenon is known as “hydra effects” (Abrams and Matsuda, 2005). Hydra effects have been recognized in many discrete-time ecological models (Sinha and Parthasarathy, 1996, Schreiber, 2003, Hilker and Westerhoff, 2006, Seno, 2008, Zipkin et al., 2009, Liz, 2010, Dattani et al., 2011, Sieber and Hilker, 2012) and continuous-time models (Abrams et al., 2003, Matsuda and Abrams, 2004) as well as models with delays (Terry and Gourley, 2010). For more details we refer to see the recent review by Abrams (2009) and the references therein on hydra effects. In this study, prey population is subject to strong Allee effects and disease, and there is no alternative food source for the predator population. Due to the limited food resource, predator population experiences intra-specific competition.

The prey–predator interaction model with disease is termed as the eco-epidemiological model, which was first introduced by Hadeler and Freedman (1989). After that, researchers are paying more interest to the eco-epidemiological models that merge the research of ecology and epidemiology (Freedman, 1990, Beltrami and Carroll, 1994, Venturino, 1995, Venturino, 2002, Beretta and Kuang, 1998, Chattopadhyay and Arino, 1999, Xiao and Chen, 2001, Chattopadhyay and Pal, 2002, Hethcote et al., 2004, Bairagi et al., 2007, Su et al., 2008). Disease transmissions are often influenced by aggregation patterns in the host population as well as its social organization. Two different types of incidence rates, i.e., density-dependent and frequency-dependent, are usually distinguished and used in epidemiology (Hethcote, 2000, McCallum et al., 2001, Begon et al., 2002, Potapov et al., 2012). Functional response is also a very important factor in the dynamical outcomes of predator–prey interaction models. Holling type I/II/III (Holling, 1959) functional responses are more common in literature.

In this article, we extend the model studied by Kang et al. (2014a) to a new model incorporating (1) the frequency-dependent disease transmission instead of the density-dependent disease transmission; and (2) the intra-specific competition in the predator population due to the limited food resource. We have performed a methodical study on the stability behavior of our proposed system to explore the interplay among the Allee effects, disease, predation and hydra effects. The rest of the paper is organized as follows: Section (2) provides the development of the model; in Section (3), we discuss the dynamics of the full model with disease free/ predation free, and compare the dynamics of submodels with and without the Allee effects. Detailed analysis of the full model and related numerical simulations are discussed in Section (4). In this section, we also provide the biological impacts of the Allee effects, disease and predation in presence of hydra effects. The paper ends with a discussion in Section (5).