Effect of hunting cooperation and fear in a predator-prey model
Introduction
Predation process assumes an essential part in advancing life evolution and maintaining ecological balance and biodiversity. Cooperation is a fundamental feature of animal social life and plays an important role in biological systems. In the context of hunting, cooperation can simply mean two or more individuals (kin or non-kin) increasing their fitness by acting together for a common goal. Predators may attack isolated or grouped prey in a cooperative and collective way. Different animals show cooperative behaviour during hunting, for instance, wild dogs (Creel and Creel, 1995), lions (Heinsohn, Packer, 1995, Stander, 1992), chimpanzees (Boesch, 1994, Boesch, 2002), birds (Hector, 1986), ants (Dejean, Leroy, Corbara, Roux, Céréghino, Orivel, Boulay, 2010, Moffett, 1988), spiders (Uetz, 1992), crocodiles (Dinets, 2015), hawks (Bednarz, 1988), and several other species (Bailey et al., 2013). There are many advantages of group hunting, such as rate of hunting success increases with the number of adults (Creel and Creel, 1995), chasing distance decreases (Creel and Creel, 1995), the probability of capturing large prey increases (Bednarz, 1988), finds food more quickly with increasing group size (Pitcher et al., 1982), helps to prevent the carcass from being stolen by other predators (Brockmann, Barnard, 1979, Vucetich, Peterson, Waite, 2004), etc. Packer and Ruttan (1988) enlisted 61 species of mammals, birds, vertebrates and invertebrates, their hunting strategy, which prey or types of prey to predate, single or multiple prey to predate at a time and percentage of their hunting success from different sources.
Predator-prey interaction is one of the most important factors in ecology. The effects of predators can be direct and lethal (Taylor, 1984), or it may be indirect and non-consumptive (Lima and Dill, 1990). Predators can impact the ecology and evolution of their prey directly by eating them, but also indirectly by influencing the behaviour of survivors (Lima, 1998). Thus, a detailed outline of fear effect (an indirect effect) is a behavioural and stress-related physiological change of prey population in the presence of predator, as prey species are always wary of possible attack. Recent experimental findings have explored that fear of predator alone can change prey’s behavior. In every taxa, all animals respond to the predation risk and show a variety of anti-predator responses, which includes habitat changes, foraging, vigilance and different physiological changes (Cresswell, 2011, Peacor, Peckarsky, Trussell, Vonesh, 2013, Preisser, Bolnick, 2008). Due to fear of predation risk, prey population can change it’s grazing zone to a safer place and sacrifice their highest intake rate areas, increase their vigilance, adjust their reproductive strategies, etc. These types of anti-predator behaviors (short-term survival strategy) are instantly beneficial by increasing the adult survivability, but it can reduce the reproduction as a long-term cost (Cresswell, 2011). Many researchers observed several predator-prey interactions, where due to fear of predation risk, the reproduction of the scared prey decreases, for example, elk-wolves (Creel, Christianson, Liley, Winnie Jr, 2007, Wirsing, Ripple, 2011), snowshoe hares-dogs (Sheriff et al., 2009) and dugongs-sharks (Wirsing and Ripple, 2011), mule deer-mountain lions (Pierce et al., 2004), etc.
In ecosystems, there are some predators who show cooperative behaviour during hunting and also create fear upon their preys. As a potential keystone species, wolves cooperate during hunting and in addition, wolves also affect their prey indirectly (Feh, Boldsukh, Tourenq, 1994, Schmidt, Mech, 1997). For example, when wolves are present, elk used anti-predator strategies and avoided areas frequented by them (Ripple and Larsen, 2000). Elk responds in the presence of wolves by altering patterns of aggregation, habitat selection, vigilance, foraging, and sensitivity to environmental conditions (Creel, Winnie Jr, 2005, Creel, Winnie Jr, Maxwell, Hamlin, Creel, 2005, Winnie Jr, Christianson, Creel, Maxwell, 2006, Winnie Jr, Creel, 2007). Lioness shows cooperative behaviours during hunting (Stander, 1992). Some of lionesses use the strategies of circled prey while other lionesses waited for prey to move towards them. Due to the fear of predation risk, zebras (Courbin et al., 2016), giraffes (Valeix, Fritz, Loveridge, Davidson, Hunt, Murindagomo, Macdonald, 2009, Valeix, Loveridge, Chamaillé-Jammes, Davidson, Murindagomo, Fritz, Macdonald, 2009) reached areas where the encounter with lioness less frequent. Thus by both killing and frightening, predators could have a dual impact upon their prey.
In mathematical modelling approach, many authors investigated impacts of hunting cooperation and fear effect in predator-prey system separately. In the context of hunting cooperation, Duarte et al. (2009) studied a three-species food chain model with hunting cooperation of the predator. Berec (2010) investigated foraging facilitation among predators in a Rosenzweig-MacArthur model. Teixeira Alves and Hilker (2017) studied a Lotka-Volterra model with hunting cooperation in predators and discussed how cooperation can affect predator-prey dynamics and also showed that hunting cooperation induces Allee effect in predators. Recently, Pal et al. (2018) investigated the impact of hunting cooperation in a discrete-time predator-prey system and observed many interesting complex dynamical behaviors. On the other hand, few researchers also investigated the impact of fear effect in the predator-prey systems with the help of mathematical modeling. In this context, Wang et al. (2016) first proposed a predator-prey model by incorporating fear of the predator on prey, where the cost of fear reduces in the birth rate of prey. They observed that fear effect can stabilize the oscillation of the system. Panday et al. (2018) investigated a three-species food chain model, by considering the growth rate of middle predator is reduced due to the cost of fear of top predator, and the growth rate of prey is reduced due to the cost of fear of middle predator. They observed that fear has the potential to stabilize a chaotic system. Recently, Pal et al. (2019) studied the impact of fear in a predator-prey model with Beddington-DeAngelis functional response and observed that the model can exhibit multiple Hopf-bifurcations.
Many researchers investigated the role of hunting cooperation (Teixeira Alves and Hilker, 2017) and the role of fear effect (Wang et al., 2016) in predator-prey systems separately. But the study on combined effect of hunting cooperation and fear in a predator-prey system is not studied yet. The aim of the present study is to investigate the impact of cooperation and fear effect in a predator-prey model simultaneously. We consider that, hunting cooperation among predators induces fear in prey population and as a result birth rate of prey population reduces. In the next section, we develop our model with hunting cooperation and fear. Basic properties such as positivity, boundedness and persistence of the model are discussed in Section 3. Equilibria and local stability analysis of the model are investigated in Section 4. Global stability of the system is investigated in Section 5. In Section 6, we perform Hopf-bifurcation analysis and discuss the stability of limit cycles. In Section 7, we analytically characterize the Bogdanov–Takens bifurcation. Numerical simulations are performed in Section 8. Finally, the paper ends with a conclusion.
Section snippets
The mathematical model
First, we consider the Lotka–Volterra predator-prey modelwhere x(t) and y(t) denote the number of prey and predator populations respectively, at any time t. Here prey grows logistically and the prey-predator interaction follows linear functional response, which is the rate at which predator captures prey. The meaning of the parameters (all non-negative) are as follows: r0 is the birth rate of the prey, d is the natural death rate of the prey, a represents the
Mathematical preliminaries
In this section, we present some basic results, such as positivity, boundedness and persistence of the system (2.2).
Equilibria & stability analysis
In this section, we analyze the system (2.2) and draw conclusions regarding the bahaviour of the solution trajectories of the system. Before proceeding to do this, we first analyze the nullclines of the system. The non-trivial nullcline of the prey species represents a curve which intersect positive x-axis at when it exists. Also the non-trivial predator nullcline of the system is a hyperbolic curve. When the birth rate of prey population is less than its death rate, the model has no
Global stability of boundary equilibrium point
Theorem 5.1 The boundary equilibrium point is globally asymptotically stable if and ρ3 > 0. Proof When and ρ3 > 0, the system (2.2) has only two equilibrium points E0(0, 0) and . The equilibrium point E0 is repelling and the equilibrium point E1 is locally asymptotically stable. Also, the system (2.2) has a bounded positively invariant region i.e. solutions starting from positive initial conditions remains positive and bounded. Hence there cannot be any periodic
Hopf-bifurcation and existence of limit cycles
Here, we explore the possibility of occurrence of Hopf-bifurcation around the interior equilibrium point E*(x*, y*) with respect to the bifurcating parameter α. In a two dimensional system Hopf-bifurcation occurs as a spiral point switches from stable to unstable (or, vice-versa) and a periodic solution appears (or, disappears). The next theorem gives the necessary and sufficient conditions for Hopf-bifurcation of the system (2.2). Theorem 6.1 The necessary and sufficient conditions for Hopf-bifurcation to
Bogdanov-Takens bifurcation
In this section, we discuss Bogdanov-Takens (BT) bifurcation of the system (2.2) in a small neighbourhood of the equilibrium point E*(x*, y*). BT bifurcation is a bifurcation of an equilibrium point in a two-parameter family of autonomous system and occurs for a system when the Jacobian matrix evaluated at the equilibrium has a zero eigenvalue of algebraic multiplicity 2. In such a situation the Jacobian matrix is similar to the Jordan block of the form . We follow the techniques and
Numerical simulations
In order to visualize the role of hunting cooperation and fear in our model (2.2), we perform extensive numerical experiments by varying the parameters α and e.
Conclusion
The foraging behaviour of animals is one of the interesting phenomena to understand the dynamics of ecological systems. There are different strategies used by both preys and predators to increase their densities. To capture prey in an efficient way, predators cooperate during hunting. Cooperative predators can hunt animals bigger or faster than themselves and the success rate of catching a prey also increases. Prey animals also show anti-predator defences as the attack rate of predators
Acknowledgments
Research work of Saheb Pal is supported by the Junior Research Fellowship from the UGC, Government of India. The authors are thankful to the anonymous reviewers and the Editor for their valuable comments and suggestions, which helped us to improve the paper.
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