Diseased prey predator model with general Holling type interactions.

Choice of interaction function is one of the most important parts for modelling a food chain. Many models have been proposed as a diseased-prey predator model with Holling type-I or type-II or type-III interactions, but there is no model with general Holling type interactions. In this paper, we study a diseased prey-predator model with general Holling type interactions. Local stability conditions of equilibrium points are derived. We obtain the permanence and impermanence conditions of the system. The conditions for global stability of the system are also derived. The system exhibits limit cycle, period-2, higher periodic oscillations and chaotic behaviour for different values of Holling parameters. One parameter bifurcation analysis is done with respect to general Holling parameters and infection rate. We utilize the MATCONT package to analyse the detailed bifurcation scenario as the two important interaction parameters are varied. It is interesting to note that a diseased system becomes a disease free system for proper choice of interaction functions. Our results give an idea for constructing a realistic food chain model through proper choice of general Holling parameters.

Introduction

Mathematical models are increasingly used to guide public health policy decisions and to control infectious disease. Epidemic dynamics is an important method of studying the spread of infectious disease qualitatively and quantitatively. The research results are helpful to predict the developing tendency of the infectious disease, to determine the key factors of the spread of infectious disease and to seek the optimum strategies of preventing and controlling the spread of infectious diseases. Mathematical models have a long history in infectious disease ecology starting with Bernoulli’s [1] modelling of smallpox and including Ross’s [2] analysis of malaria. The earliest attempt to provide a quantitative understanding of the dynamics of malaria transmission was that of Ross [2]. Ross models consisted of a few differential equations to describe changes in densities of susceptible and infected people, and susceptible and infected mosquitoes. Based on his modelling, Ross introduced the concept of a threshold density and concluded that ‘in order to counteract malaria anywhere we need not vanish Anopheles there entirely we need only to reduce their numbers below a certain figure [3]. Classical papers of mathematical modelling of infectious disease was constructed by Kermack and McKendrick (1927 [4], 1932 [5], and 1933 [6]). These papers had a major influence on the development of mathematical models for disease spread and are still relevant in many epidemic situations. Aim of ecological modelling is to understand the prevalence and distribution of a species, together with the factors that determine incidence, spread, and persistence (Anderson and May [7]; May and Anderson [8]; Bascompte and Rodriguez-Trelles [9]). Now we have models for many of the most important human emerging infectious diseases e.g., HIV (May and Anderson [10]), malaria (Aron and May [11]; Macdonald [12]), SARS-coronavirus (Anderson et al. [13]), rabies (Murray and Seward [14]), and influenza (Ferguson and Anderson [15]). Mathematical models are also being used to explore wildlife disease dynamics (Grenfell and Dobson [16]; Hudson et al. [17]) and possible routes of zoonotic disease emergence. Understanding disease dynamics across hosts is an essential first step in understanding and articulating those conditions under which new diseases can emerge from wildlife reservoirs [18]. A predator–prey system with infected prey in polluted environment is proposed by Sinha et al. [19]. Anderson and May [20] were probably the first who considered the disease factor in a predator–prey dynamics and found that the pathogen tends to destabilize the predator–prey interaction. In Rosenzweig prey-predator model, Hadeler and Freedman [21] determined a threshold above which an infected equilibrium or an infected periodic solution appear. Chattopadhyay and Arino [22] considered a three species ecoepidemiological model and studied local stability of equilibrium points, extinction criteria of species and found condition for Hopf-bifurcation in an equivalent two-dimensional model. Haque and Chattopadhyay [23] studied the role of transmissible diseases in a prey dependent predator–prey system with prey infection. Haque and Venturino reported the influence of transmissible disease in prey taking Holling–Tanner predator–prey model [24]. The dynamical behaviour of the predator–prey system was investigated when a predator avoids infected prey and the predator has alternative sources of food ([25], [26]). Bhattacharyya et al. [27] proposed a epidemiological model with nonlinear infection incidence. Das et al. [28] modified the HP model by introducing disease in the prey population. They derived conditions for population extinction and the conditions for permanent or impermanent of the system. Sahoo and Poria described a diseased prey-predator model supplying additional food to predator for biological control[29]. Recently, Haque et al. investigated predator-infected eco-epidemics systems with different functional responses[30].

In this paper, we modify the model of Das et al. [28] by introducing the general Holling type interactions in Section 2. Some preliminary results are derived in Section 3. In Section 4, the conditions for local stability of equilibrium points are derived. We derive the permanency and impermanence conditions for the model in Section 5. Section 6 presents the conditions for global stability. Section 7 contains the numerical simulation results of the model. We have done bifurcation analysis of the model with respect to general Holling parameters and have also investigated the influence of infection rate on the dynamics. The different routes of continuation of the associated bifurcations are analysed with the help of MATCONT software package ([31], [32], [33]).

Model formulation

All food chain models use some realistic interaction functions between preys and predators based on some biological hypothesis. A realistic interaction function should not allow the predators to grow arbitrarily fast, if prey is abundant. Apart from these basic biological considerations, Holling interaction functions are taken as simplest realistic interactions. Holling type-II function is defined as , where A is the maximum predation rate and K is half saturation constant. The function increase linearly if X is small. At large values of X the slope of the function decreases as predator becomes saturated but always remains non-negative. Actually, the Holling type-II function is based on the assumption that predation rate is proportional to prey density if prey is scarce. However, if the predator actively seeks out large concentration of prey the Holling type-III function is more appropriate. Since the slope of this function goes to zero for small values of X it may be suspected that the food chain will be destabilized if prey concentration becomes too small. The general Holling function is defined as , where ([34], [35], [36]). The behaviour of different types response functions are shown in Fig. 1 for different values of .

Fig. 1

Comparision of different types of Holling functional responses. (a) For some and (b) for some .

We take the following assumptions to formulate the model.

The prey population is divided into two classes, viz. (i) susceptible class whose population density is denoted by S and (ii) infected class whose population density is denoted by I. The intermediate predator whose population density is denoted by and the density of top-predator is denoted by .

A part of the susceptible prey population becomes infected at a rate , following the law of mass action.

Infected population is not in a state of reproduction and also does not compete for the resources.

Behaviour of the entire community is assumed to arise from the coupling of these interacting species, where preys on both susceptible prey and infected prey in the form of general Holling type and Holling type-I respectively. This different combinations of functional forms are taken because the capturing of infected prey is different from that of susceptible prey. Top-predator preys intermediate predator in the form of general Holling type interaction. This is in contrast to other models which assume particular Holling type interactions ([24], [25], [26]).

The infected prey population dies at the rate and the intermediate predator and top-predator die at the rate and respectively.

Under the above assumptions, we obtain the following model:

Here are respectively the susceptible prey, the infected prey, the intermediate predator and the top-predator population respectively and T is the time. The constant is the ‘intrinsic growth rate’ and the constant is the ‘carrying capacity’ of species S. and are the maximal predation rate of intermediate predator for susceptible and infected prey respectively; is the maximal predation rate of top-predator for intermediate predator; and are the half saturation constant for functional response of intermediate and the top-predator respectively; is the conversion rate of susceptible prey to intermediate predator; is the conversion rate of infected prey to intermediate predator; is the conversion rate of intermediate predator to top-predator. Here and are the general Holling parameters. From biological point of view, in real world, predators of different species may feed on preys in different types of consumption ways. For example, consider crops, aphids, and lady beetles as prey, intermediate-predator, and top-predator, respectively. In this case, it is natural to assume that the feeding type of aphids on crops is different from that of lady beetles on aphids. Thus, to describe these phenomenon, different types of functional responses are needed.

We nondimensionalize the system (1) with , , and obtain the following set of equations: The system (2) has to be analysed with the following initial conditions: ; where , .

Theoretical studies

Positive invariance

Let andwhere and . Then system (2) becomeswith . It is easy to verify that whenever choosing such that then (for i = 1, 2, 3, 4). Now any solution of the Eq. (4) with , say , is such that for all (Nagumo, M. [37]).

Boundedness

All solutions of the system (2) which initiate in are uniformly bounded.

Let us consider that .Using Eq. (2), we have predator-infected eco-epidemicssince and we get the following expression:where L  = min.since . Integrating, , C being arbitrary positive constant. Initially, when , . Therefore from the solution, we have , i.e., .

Therefore,

.

Thus, .

From the theory of differential inequality we obtain .

For , we have .

Hence all the solutions of (2) that initiate in will ultimately remain in the region , for . This proves the theorem. □

Extinction criterion

If , then . If , then . If and , then . If , then .

We haveSolving above equation we have . Hence , provided .

NowTherefore, .

.

Thus, , provided .Therefore,Therefore, ,

i.e, .

Thus, if and , then .

Now, ,

i.e., .

Thus, , provided . □

Existence and local stability of equilibrium points

The system has seven equilibrium points. is the trivial equilibrium point. The axial equilibrium point is . Disease free planar equilibrium point is , where .

The existence condition of disease free planer equilibrium point are and .

The endemic planar equilibrium point is . where . is disease free space equilibrium point where

and are the positive roots of the equation . The disease free equilibrium point exists if .

is the top-predator free equilibrium point where ,

and are the positive roots of the equation

,

The top-predator free equilibrium point exists if .

The interior equilibrium point is given by , where . . and .

The interior equilibrium point exists if , , .

The Jacobian matrix J of the system (2) at an arbitrary point is given by

The trivial equilibrium point is always unstable. The disease free planar equilibrium point is locally stable if . The endemic planar equilibrium point is locally stable if .

Since an eigenvalue associated with the Jacobian matrix at is 1, so is an unstable equilibrium point.

The Jacobian matrix at is given byFrom the Jacobian matrix , it is observed that equilibrium point is unstable if and , which are the existence condition for the equilibrium points and .

The Jacobian matrix at is given byThe characteristic roots of are and and the roots of the equation .

Hence, is stable if the conditions given in the theorem are satisfied

The Jacobian matrix at is given byThe characteristic roots of are ,  and the roots of the equation . □

Hence, the equilibrium point is stable if .

The disease free equilibrium point is locally stable if and the top-predator free equilibrium point is stable if

The Jacobian matrix at is given byThe characteristic roots of the Jacobian matrix are , and the roots of the characteristic equation is given by . where

Hence, by Routh–Hurwitz criterion [38] the equilibrium point is stable if the condition 6 holds.

The Jacobian matrix at is given byThe characteristic roots of the Jacobian matrix are , and the roots of the equation is given by . where Hence, by Routh–Hurwitz criterion [38] the equilibrium point is stable if the conditions 7 hold. □

The interior equilibrium point for the system (2) is locally asymptotically stable if the following conditions hold as follows:

,   ,   , where ’s are given in the proof of the theorem.

The Jacobian matrix at the interior point isWhere.

The characteristic equation of Jacobian matrix V is given by

. where Using the Routh–Hurwitz criteria [38] we observe that the system (2) is stable around the positive equilibrium point if the conditions stated in the theorem hold. □

Permanence and impermanence of the system

From biological point of view, permanence of a system means the survival of all species of the system in future time. Mathematically, permanence of a system means that strictly positive solutions do not have omega () limit points on the boundary of the non-negative cone. .

Let and and the following conditions are satisfied

(i) ,

(ii) ,

(iii) .

Further if there exists finite number of periodic solutions in the plane, then system (2) is uniformly persistent provided for each periodic solutions of period T.

Let be a point in the positive quadrant and be orbit through and be the omega limit set of the orbit through . Note that is bounded.

We claim that . If then by the Butler–McGehee lemma there exist a point P in where denotes the stable manifold of . Since lies in and is the space, we conclude that is unbounded, which is a contradiction.

Next , for otherwise, since is a saddle point (which follows from the existence of and ) by the Butler–McGehee lemma there exist a point P in . Now is the plane implies that an unbounded orbit lies in , a contradiction.

Next we show that . If , the condition implies that is saddle point. is the space and hence the orbits in this space emanate either or or an unbounded lies in , again a contradiction.

The condition implies that is a unstable point and also the contradiction, implies that is unstable. So by similar arguments we can show that and .

Lastly we show that no periodic orbits in the or . Let denote the closed orbit of the periodic solution in plane such that lies inside . Let the Jacobian matrix J given in (6) corresponding to is denoted by . □

Computing the fundamental matrix of linear periodic system, .

We find that its Floquet multiplier in the i direction is . Then proceeding in an analogous manner like Kumar and Freedman [39], we conclude that no lies on . Thus, lines in the positive quadrant and system (2) is persistent. Finally, since only the closed orbits and the equilibria from the omega limit set of the solutions on boundary of and system (2) is dissipative. Now using a theorem of Butler et al. [40], we conclude that system (2) is uniformly persistent.

Let and and the following conditions are satisfied and there exists no limit cycle in the plane, the system (2) is uniformly persistent.

(i) ,

(ii) ,

(iii) .

Proof of the Theorem (6) is obvious and so omitted.

Before obtaining condition for impermanence of system (2), we briefly define the impermanence of a system. Let be the population, vector, let , and is the boundary of D. is the distance in .

Let us consider the system of equation iswhere and .

The semi orbit is defined by the set where is the solution with initial value .

The above system is said to be impermanent [41] if and only if there is an such that . Thus a community is impermanent if there is at least one semi orbit which tends to boundary. □

Let and and if the condition holds, then the system (2) is impermanent.

The conditions and are obtain from existence of the equilibria points and . The given condition

implies that is a saturated equilibrium point on boundary. Hence, there exist at least one orbit in the interior that converges to the boundary (Hofbauer, [42]).

Consequently the system (2) is impermanent (Hutson and Law, [41]). □

Global stability

We have determined the conditions for global stability of interior equilibrium point through the following theorem.

The sufficient conditions for the system (2) is to be globally asymptotically stable around the equilibrium if and .

We first choose a Lyapunov function defined as follows:Calculating time derivative of the Eq. (8) along the solutions of the system (2) gives us

Thus, , provided and .

Hence the theorem folllows. □

Results

We illustrate some of the key findings using numerical simulations. We assume the parameter values , which remain unchanged for all numerical simulations. The main goal of this paper is to investigate the effects of infection rate a as well as the effects of different types of interactions for different values of and .

For and we obtain the positive interior equilibrium point . For the above set of parameter values we have and which implies that the system (2) is locally asymptotically stable around positive equilibrium .

We have shown system’s dynamics for different values of and infection rate a in Fig. 2 and Fig. 3 . From Fig. 2 and Fig. 3, we observe that the system (2) have periodic oscillations as well as chaotic bands for different general Holling parameters and infection rate. From Fig. 4, Fig. 5 , we observe limit cycle, period-2 to period-7 and chaotic dynamics. The global stability behaviour of the system (2) with different initial conditions is presented in Fig. 6 for different Holling parameter values and . Therefore, oscillatory as well as stability nature of the diseased prey population can be captured for a range of Holling parameter values. Das et al. [28] observed chaotic dynamics of the system (2) for , the period-doubling for , the limit cycle oscillation for and finally stable steady state distribution of all four species for for a particular value of and using above set of parameter values. Here we have investigated the dynamics for with different Holling interactions through bifurcation analysis.

Fig. 2

Plots of Susceptible prey, Infected prey, Intermediate predator, Top-predator vs. time in the system (2) for different values of and a.

Fig. 3

Plots of Susceptible prey, Infected prey, Intermediate predator, Top-predator vs. time in the system (2) for different values of and a.

Fig. 4

Figure depicts the Limit Cycle, Period-2, Period-3 and Period-4 behaviour of the system (2) for different values of and with infection rate .

Fig. 5

Figure depicts the Period-5, Period-6, Period-7 and Chaotic behaviour of the system (2) for different values of and with infection rate .

Fig. 6

Global stability of the system (2) with different initial conditions for (a), (b) , (c) .

Equilibrium and fold continuation

The main goal of this section is to study the pattern of bifurcation that takes place as we vary the parameters and . This is actually done by studying the change in the eigenvalue of the Jacobian matrix and also following the continuation algorithm. To start with we consider a set of fixed point initial solution, and , corresponding to a parameter set of values , most of which are taken from Hastings and Powell model [43]. The characteristics of Hopf point, the limit cycle and the general bifurcation may be explored using the software package MATCONT. This package is a collection of numerical algorithms implemented as a MATLAB toolbox for the detection, continuation and identification of limit cycles. In this package we use prediction–correction continuation algorithm based on the Moore–Penrose matrix pseudo inverse for computing the curves of equilibria, limit point (LP), along with fold bifurcation points of limit point (LP) and continuation of Hopf point (H), etc.

To start with we show in Fig. 7 (a) the continuation curve from the equilibrium point with as the free parameter. In the Fig. 7(a) we get two Hopf points (H), one limit point (LP) and two branch point (BP) of s with respect to for fixed . The first Hopf point is located at . For this Hopf point the first Lyapunov coefficient turns out to be , indicating a supercritical Hopf bifurcation. It being negative implies that a stable limit cycle bifurcates from the equilibrium when this looses stability. The branch points (BP) occur at and at . As the parameter is increasing, second Hopf point situated at . For this second Hopf point the first Lyapunov coefficient turns out to be , indicating a supercritical Hopf bifurcation. The limit point is located at with the eigenvalues as . The real part being negative, indicates that the LP is stable. The continuation curve of equilibrium point of s is also shown in same Fig. 7(a) for .

Fig. 7

Figure depicts (a) Continuation curves of equilibrium with the variation of the parameter of the variable s. (b) Hopf point continuation of the system: GH-generalized Hopf; BT-Bogdanov Takens; ZH-zero Hopf point, HH-Neutral saddle.

Now it should be recapitulated that we have started with two parameters and as bifurcation parameters. To start with we show in Fig. 7(b) the continuation curve from the Hopf (H) point with respect to . We observe six generalized Hopf (GH) points, one Bogdanov–Takens (BT) point, one zero-Hopf (ZH) point, two Neutral saddle (HH) point at different values of and . At the generalized Hopf (GH) points, where the first Lyapunov coefficient vanishes indicating that all GH points are non-degenerate, since the second Lyapunov coefficients are non-zero. The Bogdanov–Taken points are common points for the limit point curves and curves corresponding to equilibria with eigenvalues . Actually, at each BT point, the Hopf bifurcation curve (with ) turn into the neutral saddle curve (with real ). Now, we start LP point continuation from a Bogdanov–Taken (BT) point. If we choose and as free parameters and start from the BT point, the continuation curve shows two BT points and two cusp points (CP) which is shown in Fig. 8 (a). A similar analysis can also be carried out for the variables and , the results being displayed in Fig. 8(b), Fig. 9 (a) and Fig. 9(b) respectively.

Fig. 8

Continuation curves of equilibrium from limit point (LP) with the variation of the parameter of the (a) susceptible prey s and (b) Infected prey i. It depicts the H, GH, HH, CP, LP, BT, ZH etc. points.

Fig. 9

Continuation curves of equilibrium with the variation of the parameter of the (a) Intermediate predator and (b) Top-predator . It depicts the H, GH, HH, CP, LP, BT, ZH etc. points.

To proceed further we start from the Hopf point (H) in Fig. 7(a) as the initial point for with fixed , and get a family of stable limit cycles bifurcating from this Hopf point. This phenomenon is shown in Fig. 10(a), where again the Holling parameter in the system is the only free parameter. One observes that at , we have a LPC point with period 83.36766. At this situation two cycles collide and disappears. The critical cycle has a double multiplier equal to 1. From this it follows that a stable branch occurs after the LPC point. For , another LPC point occurs with one of the multiplier is greater than 1 which indicates that the cycle is unstable after LPC point. At there is a period doubling (PD) with period 92.32825, two of the multiplier is equal to 1. However, for , we observe PD again with period , one of the multiplier is greater than 1. At and , LPC’s are observed with period and respectively. For , we have a branch point cycle (BPC) with period . If we choose and period of the cycles as free parameters and start from the Hopf point (H), as shown in Fig. 7(a), then the corresponding variation of period versus is shown in Fig. 10 (b). The similar analysis is done starting from the second Hopf point (H) as initial point, as shown in Fig. 7(a) at , and we observe family of stable LPC in Fig. 11 (a). The variation of period versus is shown in Fig. 11(b). The corresponding scenario for and is exhibited in Fig. 12, Fig. 13 .

Fig. 10

(a) Family of limit cycles bifurcating from the Hopf point (H) for . (b) Period of the cycle as a function of . It depicts LPC, PD, BPC points with the variation of .

Fig. 11

(a) Family of limit cycles bifurcating from the Hopf point (H) for . (b) Period of the cycle as a function of . It depicts LPC, NS points with the variation of .

Fig. 12

Family of limit cycle bifurcating from the first Hopf point H (at ) with the variation of the parameter .

Fig. 13

Family of limit cycle bifurcating from the second Hopf point H (at ) with the variation of the parameter .

Our above analysis shows that a rich bifurcation structure exists for the predator–prey system with different Holling interactions, when and are varied over a wide range of values. It is to be noted that these two parameters represent two important physical quantities in the actual situation, respectively, biological pest control and agricultural research field. As such it may happen that such changes in behaviour may manifest in experimental studies also and so we need further extension of this studies.

Bifurcation

Bifurcation is an important tool to study the behaviour of a dynamical system. In this section, we study the dynamical behaviour of the system through bifurcation analysis with respect to and a as free parameters taking a parameter set of values .

We have done bifurcation analysis of the system (2) with respect to Holling parameter within the range , while another Holling parameter and infection rate are kept fixed. Bifurcation diagrams are presented in Fig. 14, Fig. 15 . Fig. 14(a) is the bifurcation diagram of susceptible prey of the system (2) with respect to . The Fig. 14(a) depicts chaotic bands for , periodic oscillations for and the system settles down to steady state after . The bifurcation diagram of infected prey of the the system (2) is shown in Fig. 13(b). From Fig. 14(b), it is clear that the system becomes infection free for , but the infected prey species exists for . Fig. 15(a) is the bifurcation diagram of intermediate predator () with respect to Holling parameter . The chaotic behaviour is observed in Fig. 15(a) for . Within , we observe periodic oscillations and for the system settles down to steady state. The bifurcation diagram of top-predator () with respect to is shown in Fig. 15(b). From the Fig. 14(b) it is evident that the system has chaotic bands for , periodic oscillations for and finally it settles down to steady state for .

Fig. 14

Bifurcation diagram of Susceptible and Infected prey of the system (2) with respect to taking .

Fig. 15

Bifurcation diagram of Intermediate predator () and Top-predator () of the system (2) with respect to taking .

Bifurcation analysis of the system (2) is done with respect to Holling parameter () for and infection rate with above fixed set of parameter values and it is presented in Fig. 16 . From Fig. 16(a) we observe chaotic behaviour for , steady state for and oscillatory behaviour for . The extinction scenario of the infected prey is shown in Fig. 16(b). Fig. 16(c) depicts chaotic behaviour for , stable state for and oscillatory behaviour for . The chaotic behaviour for , stable state for and extinction scenario of top-predator species for are observed in Fig. 16(d).

Fig. 16

Bifurcation diagram of Susceptible prey (s), Infected prey (i), intermediate predator () and top-predator () of the system (2) with respect to for .

One of the most important observation is that the model of Das et al. [28] with Holling type-II interaction (i.e., for ) showed chaotic behaviour, but in this model we observe periodic behaviour for with (Fig. 14 and Fig. 15). A typical period subtracting nature of the system (2) is observed. Therefore, with the increase of consumption rate of intermediate predator on prey stable coexistence of infected prey, susceptible prey, intermediate predator and top-predator is observed. For lower values of the consumption rate of susceptible prey is low and therefore the consumption rate of infected prey is very high (there will be no infected prey after some time) and as a result the system becomes disease free.

Here we have done bifurcation analysis of the system (2) with respect to infection rate a for , taking Holling parameters in Fig. 17 . We observe steady state for , periodic oscillations for and after , it shows chaotic behaviour. Therefore, for low infection rate population remain steady but with the increase of infection rate oscillatory nature become prominent. From Fig. 17(b) and Fig. 17(c), we notice that infected prey and intermediate predator have extinction possibility for , whenever susceptible prey and top-predator have no such extinction risk [Fig. 17(a), Fig. 17(d)]. A typical period adding cascade nature is observed here.

Fig. 17

Bifurcation diagram of Susceptible prey (s), Infected prey (i), intermediate predator () and top-predator () of the system (2) with respect to infection rate for .

Conclusions

A diseased food chain model with general Holling type interaction is proposed and the effects of different types of general Holling interactions are investigated. We derive sufficient conditions for local stability of equilibrium points. We also analyse the permanence and impertinence conditions of the system. The conditions for global stability are also obtained for different Holling parameters. We have explored the detailed bifurcation scenario of the proposed system varying the interaction function parameters and . The interesting outcomes are the occurrence of various kinds of bifurcation points in the process of continuation. Altogether our analysis reveals the internal complexity of the system in detailed manner. Bifurcation analysis shows that the dynamics of susceptible prey, infected prey, intermediate predator and top-predator are highly effected by the force of infection a as well as the interaction parameters and , which is in sharp contrast with the existing results [24], [25], [26]. From the simulation results, it is clear that the infected prey extinct for proper choice of interaction functions. Therefore, a diseased system reduces to a disease free system with proper choice of general Holling parameters. Therefore, we can succesfully control a disease by controlling interaction function from outside in ecosystem. We observe various types of non-unique bifurcation diagrams with respect to bifurcation parameters and a respectively, having stable fixed point, limit cycle, period-2 to higher periodic oscillations, chaotic bands etc. We notice that the infected prey may survive in the system for some range of values of general Holling parameters.

Das et al. [28] reported that rate of infection and body size of intermediate predator are prime factors for disappearance of chaotic dynamics observe in HP model. Our observations indicate that chaos disappear for suitable choice of interaction functions. The most important observation is that our model with Holling type-II interaction (for ) shows chaotic behaviour but for it shows periodic behaviour. The periodic dynamical behaviour of species was reported by many researchers from the field data [44] and laboratory data [45]. Therefore, the model with Holling type-II interactions does not always realistic in ecology, because there are lots of real food chain model which are not chaotic, but depicts oscillatory coexistence. Therefore, we conclude that a realistic food chain model depends on proper choice of general Holling parameters. Novelty of our observation is that one can control an infectious disease if the interaction functions can be controlled from outside. As research extends to higher level, we must need to continue the study for construction of real food chain model with proper general Holling interactions.