Modelling of a two prey and one predator system with switching e ect

Abstract: Prey switching strategy is adopted by a predator when they are provided with more than one prey and predator prefers to consume one prey over others. Though switching may occur due to various reasons such as scarcity of preferable prey or risk in hunting the abundant prey. In this work, we have proposed a prey-predator system with a particular type of switching functional response where a predator feeds on two types of prey but it switches from one prey to another when a particular prey population becomes lower. The ratio of consumption becomes signi cantly higher in the presence of prey switching for an increasing ratio of prey population which satis es Murdoch’s condition [15]. The analysis reveals that two prey species can coexist as a stable state in absence of predator but a single prey-predator situation cannot be a steady state. Moreover, all the population can coexist only under certain restrictions. We get bistability for a certain range of predation rate for rst prey population. Moreover, varying the mortality rate of the predator, an oscillating system can be obtained through Hopf bifurcation. Also, the predation rate for the rst prey can turn a steadystate into an oscillating system. Except for Hopf bifurcation, some other local bifurcations also have been studied here. The gures in the numerical simulation have depicted that, if there is a lesser number of one preypresent in a system, thenwith time, switching to the other prey, in fact, increases the predator population signi cantly.


Introduction
For two-prey and one-predator system, the growth rate of a predator population not only depends on the total amount of consumed prey but also on the ratio of the captured prey species. If x , x be the densities of two prey which are consumed at rates h , h by the predator respectively, then the work of Tinbergen [29] suggests that the "risk index" of the prey is denoted by r i = h i x i , (i = , ). If the rst prey x is consumed by the predator at a higher rate than the second prey x , then r is dominated by r . A predator, which is provided with more than one resource population, may try to consume particularly those prey frequently which are abundant at any time and here come the terms 'switching property of predation'. The predator prefers to consume disproportionately according to the adequacy of the prey population in case of prey switching and so, it is a frequency-dependent predation strategy. Here the predator has a choice to select between different prey species. Murdoch (1969), in his work, explained that in the case of prey switching, the relative preference r r maintains a proportionality with relative prey density x x [15]. Prey switching mainly describes a situation when predator prefers a particular prey over others due to their abundance and availability in the environment, i.e., in some cases, there exists a strong preference for those prey which can be easily found and a weak preference for those prey which are available in a smaller quantity. This preference is obtained from the relationship between ratio of prey available in the system and ratio of prey consumed by the predator which is de ned as: h h = c x x ; where x , x respectively be the biomass of rst and second prey population which are consumed at rates h , h by the predator. Here c is the term which represents the preference of the predator for rst prey species (x ), i.e., c > implies predator's preference for rst prey and c < implies predator's preference for second prey. So, if c starts to increase with the increase of x x , it indicates that the predator switches to x with time. On the other hand, if predator searches for the prey which is present in lesser quantity in the system, then c decreases with an increase of x x and the opposite prey switching occurs. Now, prey switching may occur due to di erent reasons: in a system, if a predator always consumes the easily available prey, then with time, that prey population decrease and the predator, on that situation, have to look for other prey; secondly, those prey whose abundance is higher can be riskier to hunt.
Switching of prey increases the predator's ability to consume more than one prey instead of sticking up to one particular prey and it increases their growth with a higher rate too. The occurrence of switching behaviour sometimes depends on the environment as well as on the foraging timing. The snail Acanthina in rocky shores of Southern California mainly hunt two types of prey named mussel (Mytilus edulis) and barnacle (Balanus glandula). , in their experiment, has concluded that switching occurs for the snails when the prey ratios are varied and also the patches of the abundant prey is provided before experiment [19]. So, prey switching occurs for the higher prey density in this case. On the other hand, in an experiment [17], Guppies have been taken as predator and it has been observed that they hunt for Tubi cids and Drosophila on the bottom and at the top of an aquarium respectively. The authors have concluded here that the switching behaviour occurs when the prey density is lower and the predator tries to consume more when the prey is scarce and if both prey species are scarce, then they hunt according to their adequacy. Next, let us consider the case for stone ies. Elliott (2004), in his experiment, has shown that the mature larvae of Dinocras cephalotes and Perla bipunctata do not show any switching behaviour but the larvae of Isoperla grammatica and Perlodes microcephalus prefer to consume Baetis rhodani over Chironomus sp. when the abundance of Baetis is higher in the system [5]. Some literatures reveal that predator can switch the prey population at the time of consumption due to the formation of a search image [9]. It has been observed through experiment that Bombus pensylvanicus exhibits a kind of switching behaviour due to formation of search images [21]. Moreover, in the case when a predator is hunting a particular prey species for a long time, then they become e cient to hunt more of those prey and this leads to the occurrence of prey switching. This is observed in the experiment of Bergelson (1985): the dragon y Anax junius always go for either may y nymphs or tubifex worms according to the prey abundance, i.e., the predator always try to hunt those prey which they have already captured before [3]. Switching of prey population mainly helps a system to become stable and in fact, there are some results show that a system where two prey population coexist becomes stable in the presence of prey switching [1]. One reason for this situation is that predator starts searching for another resource when a particular prey population becomes very low and rare and this can create prey refugia and that prey population can save themselves from the verge of extinction [7].
There are some research works reveal that switching e ect make a signi cant impact on the stability of the prey species [16,25,27]. Comins &Hassell in 1976 andTeramoto et al. in 1979 studied two prey-one predator species where the prey species involve in intra-speci c competition [4,28]. According to their results, the competing prey species can coexist in the presence of switching e ect. Matsuda et al. (1986), in their work, proposed two-trophic and three-trophic systems to analyze the e ect of predator switching on stability [11]. Baalen et al. (2001), in their work, have considered that the predator population switch to constant alternative food when the preferred prey is scarce [30]. They have concluded that switching to alternative food does not stabilize the equilibrium but promote persistence. Further Siddon and Witman, in 2000, performed two years duration eld caging experiments on the Isles of Shoals in Maine (USA) to analyse behaviorally mediated indirect e ects in shallow subtidal food-web [26]. They observed the predation of crab (Cancer borealis) on sea urchins (Strongylocentrotus droebachiensis) in three di erent habitats named codium fragile algal beds, barrens and mussel beds to analyse the impact of prey switching. They concluded that crabs consume more urchins in the codium and barren habitats than the mussel habitat. In the mussel habitat, crabs start to feed on mussels instead of urchins and here the prey switching occurs. On the other hand, in barren habitat, crabs consume more urchins which indirectly increase the density of Diplosoma spongiforme from Ascidiacea class. TANSKY (1978), in their work, have considered a general switching function in such a manner that the 'risk index' of prey increases when its density increases and decreases when the other prey population increases [27]. So, the prey switching occurs according to the prey abundance in a system, i.e., ∂r ∂x > , ∂r ∂x < and ∂r ∂x < , ∂r ∂x > . They have concluded that the switching towards one prey becomes stronger when that prey intensity becomes adequate in the system. There are some literatures consist of general switching functional response which considers that the predator fractionally allocates the searching e orts for each prey [2,11].
The complexity of any model mainly depends on the choice of functional responses. Though Lotka and Volterra rst incorporated the prey-predator interaction, later several works have been done by re ning the model to observe di erent dynamical behaviour [12,14,22,23,24]. In this work, we have shown that the inclusion of some relevant biological arguments has made the model system more realistic. A prey-predator interaction, where the predator's preference over the food is considered, shows more biologically relevant properties as the extinction of predator does not fully depend on the extinction of a particular prey species. So, in this literature prey-predator interaction with switching behaviour of predator is proposed to observe its impact on the dynamics. Here, both the prey are engaged in intra-speci c competition and grow with respective logistic growth rate. In the following section, two prey -one predator system is formulated with prey switching, i.e., where the predator chooses the prey population for consumption depending on their abundance in the system. Section 3 shows that the proposed system is biologically well-posed as both prey and predator species are positive and bounded with time. Extinction criterion of the predator is stated in Section 4 and the equilibria with the feasibility criterion are described in Section 5. Local stability analysis and persistence of the system are discussed in Sections 6 and 7 respectively. The conditions for which equilibria change their stability through bifurcations are stated in Section 8. The global stability criteria are discussed in Section 9 and moreover, the numerical simulation in Section 10 support the analytical ndings. The work is ended with a brief conclusion about the ndings.

Mathematical Model: Basic Equations
An ecological system in terms of mathematical equations mainly provides the proper insight into the system and it becomes easier to nd the dynamics of the model with time. In this work, a prey-predator interaction is considered with two prey species where the predator population exhibits switching behaviour when a prey quantity becomes smaller. The complex and rich dynamics induced by the switching e ect is mainly highlighted in this work. The proposed system consists of two prey (x , x ) and one predator species (y). It is assumed that in absence of predator population, rst prey population (x ) grows according to logistic law with intrinsic growth rate r and carrying capacity K and the second prey species (x ) grows logistically with intrinsic growth rate r and carrying capacity K . The parameters η and η represent the consumption rate coe cients of the predator on rst and second prey species respectively. The predator is assumed to be specialist one as they live depending on only these particular two prey species. In that case, let, c and c be the biomass conversion rates for rst and second prey species respectively. The parameter d denotes the mortality rate coe cient of the predator population. It is later observed that such switching e ect helps to increase the growth of the predator population. First we consider Lotka-Volterra equations with two prey and one predator system in (2.1): In system (2.1), the consumption rate of predator for rst (or, second) prey depends on that particular prey only. In Holling type functional responses, the consumption rate coe cient is a single parameter (η or η ) actually. Now we consider a situation when predator searches for second prey in scarcity of rst prey and vice-versa. In this case, let us replace η by γ x + x + x and η by γ x + x + x . When x is much smaller than x , i.e., x x , then + x + x x becomes a large quantity which implies η → for very small x (compare to x ). So, in the scarcity of x , predator switches to x population for consumption. On the contrary, when x is much smaller than x , i.e., x x , then + x + x x becomes a large quantity which implies η → for very small x (compare to x ). So, when there is a very small amount of second prey population (x ) exists in the system, the predator switches to rst prey population (x ) for consumption. As a consequence, the predator population switches to that prey which is found to be adequate in the system. This is a kind of density dependence predatory rate which diminishes at low prey density. This may occurs when the prey population becomes rare and the predator switches the predation to another prey species and also the individuals of small prey population nd enough hiding places. Considering the switching e ect of the predator, system (2.1) becomes: where the model parameters r , r , K , K , γ , γ , c , c , d are positive constants with < c , c < . If we consider h and h be the predation rates of the predator for rst and second prey species respectively, then the ratio of the predation rates in absence and in presence of switching e ect become . Figure 1 shows that the ratio of consumption rate is much higher when switching e ect is considered in the system that helps to grow predator population.

Positivity and Boundedness
We check positivity and boundedness of the variables of system (2.2) which guarantee the well-posedness.
Proof. Right hand sides of systems (2.2) are continuous and locally Lipschitzian on C implies (x (t), x (t), y(t)) of (2.2) exists and is unique on the interval [ , τ), where < τ ≤ +∞ [8]. From the rst equation of (2.2), we get Also, the second equation of (2.2) gives The last equation of system (2.2) gives Theorem 3.2. Solutions of system ( . ) starting in R + remain uniformly bounded for t > .
Proof. From the rst equation of (2.2): Further from the second equation, As Therefore, all solutions of system (2.2) enter into the region:

Extinction Scenarios
Following theorem provides the criterion for which predator population goes extinct from the system in long run.
Proof. For large time t: Remark. If the overall growth rate of predator fails to exceeds its mortality rate, then with time, the predator population goes extinct from the system.

Planar Equilibrium Points
with P = r γ K − r γ K and x * is the positive root of the equation:

Local Stability Analysis
Local stability conditions of the equilibrium points can be determined by the eigenvalues of the corresponding Jacobian matrices. Now, the Jacobian matrix of system (2.2) is given by So, λ = r > , λ = r > and λ = −d < .
Theorem 6.1. E is always a saddle point.
So, λ = r > , λ = −r < and λ = c γ K +K − d. So, we have the following theorem: So, we have the following theorem: . So, one eigenvalue is λ = a = r > and other two eigenvalues are roots of the equation: λ − a λ − a a = . So, two eigenvalues have negative real parts but one eigenvalue is always positive.
Theorem 6.5. E is always a saddle point.
. So, one eigenvalue is λ = a = r > and other two eigenvalues are roots of the equation: λ − a λ − a a = . So, two eigenvalues have negative real parts but one eigenvalue is always positive. Theorem 6.6. E is always a saddle point.
Characteristic equation for E * (x * , x * , y * ) is where P = −(a + a ) > , P = a a − a a − a a − a a and P = a (a a − a a ) + a (a a − a a ). Consider, ∆ = P P − P , and so by Routh-Hurwitz criterion all the roots of equation (6.1) have negative real parts if P > and ∆ > . Hence, we have the following theorem: Theorem 6.7. Interior equilibrium point E * (x * , x * , y * ) is locally asymptotically stable (LAS) if P > along with P P > P .

Persistence
In the ecological context, persistence means the long term survival of all species which exist initially.
Proof. Let the average Lyapunov function be V(x , x , y) = x θ x θ y θ where θ i for i = , , are positive. In the interior of R + , we havė If the system is persistent, then ϕ(x , x , y) > for all boundary equilibria of the system. The values of ϕ(x , x , y) at the boundary equilibria E , E , E , E , E and E are as follows: is positive at the boundary equilibria E , E , E , E , E and E for some positive θ i , i = , , . Hence system (2.2) is persistent [6,13] Remark: This condition guarantees the instability of the boundary equilibria of system (2.2).

Bifurcation Analysis
Change of stability around the equilibria have been discussed in this section. We have used Sotomayor's Theorem [20] and the Hopf Bifurcation Theorem [18] to observe the local bifurcation analysis of system (2.2). In order to apply Sotomayor's Theorem, the Jacobian matrix at the bifurcating equilibrium point has to contain one zero eigenvalue. Let V = (v , v , v ) T and W = (w , w , w ) T be the eigenvectors of J| (eq. point) and J| T (eq. point) respectively corresponding to zero eigenvalue at the equilibrium point.
Proof. [TC] be the value of d such that J| E has a simple zero eigenvalue at d = d [TC] . So, at d = d [TC] : Here, λ = −r < , λ = −r < and λ = .
By Sotomayor's Theorem, system (2.2) undergoes a transcritical bifurcation around E at d = d [TC] . . Let, dx denote the slope of g (x , x , y) = and dy (g ) dx , dy (g ) dx denote the slope of g (x , x , y) = .
Also, dy (  has a simple zero eigenvalue. Here, V = (−a a , a a , a a − a a ) T and W = (−a a , a a , a a − a a ) T .
Hopf Bifurcation at E * (x * , x * , y * ) Here, let us consider γ as bifurcation parameter to check the instability of the equilibrium point E * . The characteristic equation of system (2.2) at E * (x * , x * , y * ) is where P = −(a +a ), P = a a −a a −a a −a a and P = (a a −a a )a +a (a a − a a ).
Comparing real and imaginary parts: A co-dimensional 2 bifurcation is a cusp bifurcation where the critical equilibrium point has a zero eigenvalue and the quadratic coe cient for the saddle-node bifurcation vanishes. Two branches of saddle-node bifurcation curve or a saddle-node bifurcation curve and one transcritical bifurcation curve meet tangentially at the cusp bifurcation point forming a semi-cubic parabola. Moreover, in a two parametric plane a local bifurcation of co-dimension two occurs when Hopf-bifurcation curve and saddle-node curve intersect each other and this bifurcation is called Bogdanov-Takens bifurcation. Hence system (2.2) undergoes a co-dimensional two bifurcation which is known as cusp bifurcation at (r , γ ) ≡ (r [CP] , γ [CP] ) and the point (r [CP] , γ [CP] ) in the r -γ parametric plane is the intersection of the two branches of saddle-node bifurcation, where they meet each other tangentially.
Proof. From the rst equation of system (2.2), we have From the second equation of system (2.2), we have Therefore, in the limit x (t) and x (t) are given by solutions of the equations: . Hence proved.

Remark:
In system (2.2): global asymptotic stability of E (K , K , ) guarantees that it is locally asymptotically stable.
Theorem 9.2. The interior equilibrium point E * is globally asymptotically stable in the region: Proof. Consider an appropriate Lyapunov function: Here V(x , x , y) is a positive de nite function for all (x , x , y) except at (x * , x * , y * ). The time derivative of V computed along the solutions of system (2.2) is given bẏ . Parametric values have been taken from Table 1.
So,V < when the stated conditions are satis ed in Ψ . AlsoV = when (x , x , y) = (x * , x * , y * ). HenceV is negative de nite under some parametric restrictions and LaSalle theorem [10] implies global asymptotic stability of E * .

Numerical Simulation
Numerical simulation is a vital part for any model system as this support the analytical results. In this work, we have considered a two-prey and one-predator system where the predator population switches between two prey population according to their abundance. We have considered ve sets of parameters to analyze the local dynamics of the system. To draw Figure 2 and Figure 3, we set parametric values in Table-1. Figure 2 shows that the trajectory approaches (for d = .
) to the planer equilibrium point E ( , , ) where two prey species coexist in a steady state. But if the predator mortality rate starts to decrease, then below a certain value (d [TC] ) the trajectory approaches towards interior equilibrium point E * instead of E . Figure 3 shows that the trajectory approaches (for d = .
) to E * ( . , . , . ) where all the species coexist in a steady state. So, from the steady state of E if d goes below a threshold value, then E switches from stable to unstable state and the system undergoes a transcritical bifurcation around E at d = d [TC] = . (see Figure  4).
Next we consider another set of values of the parameters as mentioned in Table 2 Figure (6.a) and (7.a) shows that for γ = . , the system approaches to interior steady state E * ( . , . , . ). After crossing γ [H ] = . , gure (6.b) and (7.b) depict an oscillating behaviour of the system and occurrence of a stable limit cycle around unstable equilibrium  . Parametric values have been taken from Table 1.  Table 1.   Table  2. > γ [H ] . Parametric values have been taken from Table 2. point E * ( . , . , . ) (as l < ) for γ = . . After crossing γ [H ] = . the limit cycle disappears and gure (6.c) and (7.c) show that for γ = .
> γ [H ] , the system again converges to stable interior equilibrium point E * ( . , . , . ). Hence, system (2.2) undergoes two hopf bifurcations at γ = γ [H ] and γ = γ [H ] respectively. Figure 8 exhibits the hopf bifurcation diagram taking γ as the bifurcation parameter and we can conclude that the consumption rate of predator for the rst prey population can stabilize as well as destabilize the system. From gure (6.b) and (7.b), it is shown that for γ = . and d = , the system performs oscillation and a stable limit cycle occurs around the unstable equilibrium point ( . , . , .
). But decreasing value of d can settle the system into stable state through Hopf bifurcation. Figure 9 shows that for d = . , the system tends to stable interior equilibrium point E * ( . , . , . ). Figure 10 portraits the Hopf bifurcation diagram taking d as the bifurcation parameter and so, the system undergoes a supercritical Hopf bifurcation at d = d [H] = .
). To draw the one-dimensional saddle-node bifurcation and two-dimensional BT and cusp bifurcations, we have considered another set of parametric values mentioned in Table 3. From the coe cients of equation (5.1), it is observed that the system has either one or three equilibrium points depending on the choice of parametric values. Figure 11 shows that the system undergoes two non-degenerate saddle-node bifurcations taking r as the bifurcating parameter which are denoted by LP and LP where LP : (E * ; r [LP ] ) ≡ ( . , . , . ; . ) with normal form coe cient a = − . e − and LP : < γ [H ] , (7.b) Occurrence of limit cycle for γ = . ∈ (γ [H ] , γ [H ] ) and (7.c) limit cycle disappears for γ = .
> γ [H ] . Parametric values have been taken from Table 2. < d [H] . Parametric values have been taken from Table 2.  . ) with normal form coe cient a = − . e − . From the calculations and also from Figure 11, it is observed that when r lies below r [LP ] and above r [LP ] , there exists only one stable interior equilibrium point but when r lies between (r [LP ] , r [LP ] ), then there are three equilibria out of which two equilibria are stable and one is a saddle point. So, a case of bistability has occur in this case. Bistability is a phenomenon which indicates that the trajectories can converge to two di erent equilibrium points for same parametric values depending on the choice of initial points. Figure 12 shows that for r = .
> r [LP ] , there is again only one stable equilibrium point E * ( . , . , . ). Figure 13 and Figure 14 show the occurrence of two-dimensional bifurcations along with one dimensional local bifurcations. Figure 13 shows that between LP and LP , there is a point where the two saddle-node bifurcation curves meet tangentially forming a cusp bifurcation denoted by "CP".  Table 3. Figure (11.b) is the enlarged version of Figure (11.a).  Table 3.  Table 3. Figure (13.b) is the enlarged version of Figure (13.a).  Table 3. Another set of parametric values has been considered in Table 4 to analyse the count of predator population for increasing intrinsic growth rate of rst prey population (r ) when the growth rate of second prey species (r ) is smaller than the rst prey. Figure 15 shows that when the consumption rate of predator for rst prey population (γ ) increases, the predator population increases for increasing r . In this case, if the predator consumes the rst prey at a lesser quantity and chooses the second prey frequently, then the growth of predator becomes much smaller than the other cases. On the other hand, if the predator consumes the rst prey at a higher rate, then their growth becomes higher enough due to the adequacy of rst prey.
Further the parametric values mentioned in Table 5 are chosen to analyse the count of predator population for increasing intrinsic growth rate of second prey population (r ) when the growth rate of rst prey species (r ) is smaller than the second prey. Figure 16 shows that when the consumption rate of predator for second prey population (γ ) increases, the predator population increases for increasing r . In this case, if   Table 4.  the predator consumes the second prey at a lesser quantity and chooses the rst prey for hunting, then the growth of predator become smaller. On the other hand, because the system has a su cient number of second prey, if the predator consumes more of the second prey, their growth becomes higher. So, adopting switching strategy based on adequacy of prey population increases the overall growth of the predator population.

Discussion
Switching of prey population is a kind of consumption strategy of predator which is frequency-dependent, i.e., the predator prefers to consume that prey which presents in the system with su cient amount. Prey switching can be applied when the predator has a choice between two or more prey species. Oaten & Murdoch have made an experiment on snails to observe their switching behaviour on prey species mussel and barnacle [19]. The snails prefer to hunt the prey with higher density. Taking Tubi cids and Drosophila as the prey species, switching behaviour is observed in Guppies when one prey is available in lesser amount in environment [17]. For the case for stone ies, Elliott has shown that the mature larvae of Isoperla grammatica and Perlodes microcephalus prefer to consume Baetis rhodani over Chironomus sp. when the abundance of Baetis is higher in the system [5]. We have formulated the underlying model to observe how this prey switching behavior makes an impact on the growth of the predator population.
In this work, two prey-one predator system has been considered where the predator population chooses the prey population for consumption according to their abundance. Both the prey population here grow with logistic law. The predator is considered here as a specialist one as the growth rate of predator depends only on the prey population x and x . The proposed model is biologically well-de ned as the system variables are positive and bounded for all time. The extinction criterion shows that if the overall growth rate of predator fails to exceed the mortality rate, then with time, the predator will go extinct from the system. Stability analysis shows that the planer equilibrium point containing both prey population and also the coexisting equilibrium points are asymptotically stable under some parametric restrictions. The mortality rate of predator (d) plays a crucial role to control the system dynamics. The predator-free equilibrium changes from unstable to a stable position through transcritical bifurcation when d crosses the bifurcation threshold. Further, the consumption rate of predator for rst prey species (γ ) also plays an important role. It is observed in the numerical simulation that, γ has a stabilizing as well as destabilizing e ect. It means an increasing value of γ can turn a stable system into an oscillating system through Hopf bifurcation but for further increment, the system recovers its stability through another Hopf bifurcation and converges to steady-state. Instead of γ , regulating the predator mortality rate coe cient (d) can lead to a system with oscillation. The system can exhibit two saddle-node bifurcations by regulating the intrinsic growth rate of rst prey population (r ). Moreover, bistable behaviour is also observed in the system in a certain range of r . In the range (r [LP ] , r [LP ] ), there are three interior equilibrium points out of which two points are locally asymptotically stable and one is a saddle point. So, a trajectory tends to either of those stable equilibria depending on the choice of the initial point. One of the bene ts of prey switching is that the growth rate of predator always increases as they can choose that particular prey which is abundant in the system. If the intrinsic growth rate of second prey population is lesser than the rst prey, then Figure 15 reveals that when the consumption rate for rst prey lies below the predatory rate of second prey, the predator population can not increase that much but when the consumption rate for rst prey becomes higher than the consumption rate for the second prey, then the count of predator population becomes signi cantly higher. Also, when the intrinsic growth rate of the rst prey population is lower than the second prey, Figure 16 depicts that the predator population increases with a higher rate for a higher predatory rate of the second prey. So, instead of searching the prey which is inadequate in the system, the predator should consume that prey that is present in ample amount as it bene ts their growth. The overall analysis reveals that the switching e ect helps to increase the predator population and as the switching depends on the adequacy of prey species, so, the prey species can save themselves from the verge of extinction.
The overall analysis shows that the proposed model exhibits interesting results but this model can be re ned further. The consumption term here is dependent only on the prey population but one may consider another consumption term which is dependent on the predator population too. Moreover, incorporation of 'gestation delay' can also be considered as the consumption procedure is not instantaneous and the predator needs some time to digest the food before further hunting. Also, the incorporation of the Allee e ect can make the work more realistic. The future model can be formulated considering all these facts to make it more realistic.