An eco-epidemiological model with predator switching behavior

Deepak Tripathi; Anuraj Singh

doi:10.1515/cmb-2023-0101

Open Access Published by De Gruyter Open Access November 3, 2023

An eco-epidemiological model with predator switching behavior

Deepak Tripathi and Anuraj Singh

From the journal Computational and Mathematical Biophysics

https://doi.org/10.1515/cmb-2023-0101

Abstract

Switching mechanism is adopted by predator populations when they are provided with two types of prey: susceptible and infected. In this study, we propose a modification of an eco-epidemiological model with the predator switching mechanism. In the presence of switching behavior, the existence of steady states and their stability have been discussed. The qualitative changes in the proposed model have been observed by the existence of transcritical and Hopf bifurcation. Numerical simulations are performed to support our numerical findings. In the context of species’ survival when disease is present in the system, it gives some theoretical views for eco-managers to understand the dynamics.

Keywords: susceptible prey; infected prey; predator switching behavior; extinction; Hopf bifurcation

MSC 2010: 34D20; 92B05; 92D25

1 Introduction

Mathematical biology points out that prey-predator modeling has a crucial role in scientific research. In ecosystem, at the time of interactions of different species, various kinds of diseases may take place and spread among the species. Theoretical research and field observations have also determined the widespread presence of different microparasite and infectious diseases. Epidemiology and ecology are two different areas over the years. However, as time passed, these two areas become very close to each other and a new branch of study comes in picture, known as eco-epidemiology. In 1986, Anderson and May [1] first observed the dynamics of eco-epidemiology model in which the interactions between predator and infected prey take place. Recently, many researchers [2,4–8,11,29,30] have shown their interest to study the mathematical modeling of infection-dominated eco-epidemiological populations. Haque and Chattopadhyay [10] studied the role of transmissible disease in the prey-predator system having infection in prey. The effect of infection on the stability of a prey-predator system with different functional response was studied by Bairagi et al. [2]. Han et al. [9] studied eco-epidemiological models for susceptible-infected-susceptible and susceptible-infected-recovered diseases with standard mass action incidents.

Recently, the article on eco-epidemiology has already secured that predators have a deep inclination to consume a unreasonable amount of infected prey [28]. Usually, parasite infection produces a switch in the behavioral form of the prey that makes them more exposed for predation. Parasite infection makes the species weaker, that is why, they live in locations that are easily accessible for predators. It could be observed in case of aquatic snails and infected fish living near the surface of water. Moreover, due to infection, the infected prey become less active so they are easily grabbed by the predators [19]. Peterson and Page [24] investigated that when moose are infected with lungworm, the encounter rate of wolf is more successful in Isle Royale in Lake Superior. The less active infected prey are targeted by predators such as lions and wolves [18,26]. Lafferty and Morris [14] carried out an experiment and observed that the predation rates of piscivorous birds on infected fish are much higher than those on susceptible fish.

The aforementioned observation and tendency of favorable predation would cause basic reduction in the biomass of infected prey due to which predators forced to move their concentration toward susceptible prey for the moment [20]. This tendency of changing the predation priority is known as predator switching behavior. In switching behavior, rare species become the most abundant one. Lawton et al. [15] documented instances of switching among the aquatic invertebrate predator species, Notonecta and Ischnura, in controlled laboratory settings. They concluded that switching is a typical aspect of predator behavior. Additional instances of switching can be found in [22]. When the population of a prey species become rarer in comparison with another prey species, the rate at which individual prey members are targeted and attacked by predator diminishes. Regarding the predator switching phenomenon, a lot of studies have been carried out [12,13,21,22,27]. Bhattacharyya and Mukhopadhyay [21] observed the impact of predator switching with and without prey group defense under non-homogeneous mixing of population. Saha and Samanta [25] also investigated a two prey and one predator system with the predator switching mechanism and noted that it influences the Hopf bifurcation of the system as different ecosystem parameters are varied.

The motivation for the model presented in this article stems from the Serengeti ecosystem that exhibits a division into two disjoint habitats that support diverse wildlife populations: (a) open southern grasslands with low rainfall that support a relatively low biomass of short growing grasses and (b) wooded northern grasslands with higher rainfall that support tall, highly lignified grasses [3,16,17]. Rainfall plays a fundamental role in influencing primary productivity within both types of grasslands [3,16,17]. Annually, approximately one million wild beast undertake a migration across the Serengeti Mara ecosystem [13]. As the wet season commences, they move from the rivers to inhabit the woodlands, primarily consuming fresh green grass leaves along with various herbs. Many of these plants could potentially be annuals. With the maturation of grasses and cessation of rainfall, their dietary habits gradually incorporate an increasing amount of browse [13]. Consequently, they relocate their self to plant communities where browse is more abundant. In this study, we formulate a mathematical prey-predator model in which prey suffers with a micro-parasite infection. The interesting part of this study is predator switching strategy and due to this how the stability and qualitative behavior of the system affected.

This article is organized in the following manner: Section 2 contains the model formulation of the proposed model system. The positivity and boundedness of the solution is presented in Section 3. Section 4 stands for steady states of the proposed model system. Section 5 represents the local stability analysis of existing steady states. Bifurcation analysis is discussed in Section 6. Numerical simulations and conclusion are presented in Sections 7 and 8, respectively.

2 Model formulation

This section contains the assumption and mathematical formulation of the prey-predator model. To develop the model, the following assumptions have been considered:

When there is no infection and predation, the prey population grows logistically.
When the microparasite infection appears in prey, the prey populations are split into two separated classes as susceptible ( x ) and infected ( y ) .
It is considered that only susceptible populations are able to reproduce with logistic rule and infected populations, die before having the capability of reproducing. However, the infected population ( y ) still contributes with x to population growth toward the carrying capacity.
The mode of infection spreading follows the law of mass action. The disease is propagating among the prey population only, and the disease is not genetically inherited. Recovery is not possible for the infected prey.
Switching mechanism of predators is incorporated in the model by discussing the functions that depict the switching mechanism.

Considering the aforesaid assumptions, an eco-epidemiological model has been proposed in the following:

(1) d x d t = r x 1 − ( x + y ) k − λ x y − α 1 x z 1 + y x 2 = f ( x , y , z ) , d y d t = λ x y − α 2 y z 1 + x y 2 − μ y = g ( x , y , z ) , d z d t = β α 1 x z 1 + y x 2 + β α 2 y z 1 + x y 2 − δ z = h ( x , y , z ) .

System (1) is analyzed with the initial restrictions x ( 0 ) > 0 , y ( 0 ) > 0 , and z ( 0 ) > 0 .

In model System (1), the interaction functions α 1 x z 1 + y x 2 and α 2 y z 1 + x y 2 mathematically describe the switching behavior [27] with predation rates α 1 and α 2 , respectively. Here, β is the conversion coefficient and its value lies, β ∈ ( 0 , 1 ] . From a biological point of view, these functions explain that the predation rate on a species decreases when the population density of that species becomes scarce compared to that of other species. The ecological meaning of the remaining ecosystem parameters and variables is mentioned in Table 1.

Table 1

Ecological interpretation of parameters and variables involved in the model System (1)

Parameters and variables	Ecological meaning	Unit
x	Biomass of susceptible prey species	[Density]
y	Biomass of infected prey species	[Density]
z	Biomass of predator species	[Density]
r	Intrinsic growth rate of prey	[ Time ] ‒ 1
k	Carrying capacity of prey	[Density]
λ	Infectious contact rate	[ Time ] ‒ 1
μ	Mortality rate of infected prey	[ Time ] ‒ 1
δ	Mortality rate of predators	[ Time ] ‒ 1
α 1 , α 2	Predation rate	[ Time ] ‒ 1
β	Conversion coefficient	—

3 Positivity and boundedness of solutions

Theorem 3.1

All the solutions x ( t ) , y ( t ) , and z ( t ) of the aforementioned System (1) with the initial conditions x ( 0 ) > 0 , y ( 0 ) > 0 , and z ( 0 ) > 0 are positive and uniformly bounded for all t ≥ 0 .

Proof

Clearly, the interaction terms of model System (1) are continuous and have partial derivatives on the region R + 3 = { ( x , y , z ) ∈ R + 3 : x ( 0 ) > 0 , y ( 0 ) > 0 , z ( 0 ) > 0 } . Therefore, these terms satisfy the Lipschitz condition on R 3 . Thus, the solution of model System (1) exists and unique.

Let us define u = x + y + z and then, we obtain

(2) d u d t = d x d t + d y d t + d z d t ,

Using the model System (1) and equation (2), we have

d u d t ≤ r x − μ y − δ z .

As r is the intrinsic growth rate of susceptible prey and k is the carrying capacity of prey, then it could be concluded that x is bounded above by k ¯ = max { x ( 0 ) , k } and also consider d ¯ = min { μ , δ } and then we obtain

d u d t ≤ ( r + d ¯ ) k ¯ − d ¯ u .

For long run, i.e., as t → ∞ , we have 0 ≤ u ≤ k ¯ ( r + d ¯ ) d ¯ .□

3.1 Extinction scenarios

d z d t = β α 1 x z 1 + y x 2 + β α 2 y z 1 + x y 2 − δ z , d z d t ≤ z ( β α 1 x 3 + β α 2 y 3 − δ ) , d z d t ≤ z ( β ( α 1 + α 2 ) − δ ) , d z d t ≤ − η z ,

where, η = ( δ − β ( α 1 + α 2 ) ) . Hence, lim t → ∞ z ( t ) = 0 . On behalf of this finding, we could summarize the following.

Theorem 3.2

If the overall growth of predator is greater than its mortality rate, then with time, predator population goes extinct from System (1).

4 Steady states

We detect the steady states of model System (1) by equating f ( x , y , z ) , g ( x , y , z ) , and h ( x , y , z ) to zero and solving the algebraic equations. Thus, we have the following four types of possible steady states:

Infection and predator-free steady state E 1 ( k , 0 , 0 ) always exists. In this state, prey biomass survive with the strength of carrying capacity.
The infection-free steady state E 2 δ α 1 β , 0 , r ( k α 1 β − δ ) k α 1 2 β exists provided k > δ α 1 β . That is, if the carrying capacity of prey is greater or equal to the ratio of mortality rate of predators to biomass conversion rate of predators, then the infection-free state is possible.
If the infectious contact rate is higher or equal to the ratio of mortality rate of infected prey to carrying capacity of prey, i.e., λ > μ k , then the predator-free steady state E 3 μ λ , r ( k λ − μ ) λ ( r + k λ ) , 0 exists.
The coexisting steady state E * ( x * , y * , z * ) is the positive solution of r 1 − ( x + y ) k − λ y − α 1 z 1 + y x 2 = 0 , λ x − α 2 z 1 + x y 2 − μ = 0 and β α 1 x 1 + y x 2 + β α 2 y 1 + x y 2 − δ = 0 .

5 Stability analysis

In this section, we discuss the local stability analysis of the existing steady states. Consider a small amount of disturbance from the steady state, and by linearizing the resulting equation, we have the following Jacobian matrix:

(i) At E 1 ( k , 0 , 0 ) , the calculated Jacobian matrix at E 1 is

J E 1 = − r − ( r + k λ ) − α 1 k 0 ( k λ − μ ) 0 0 0 − δ .

The eigenvalues of matrix J E 1 are λ 1 = − r < 0 , λ 2 = ( k λ − μ ) , and λ 3 = − δ < 0 . Using these eigenvalues, the stability of steady state E 1 could be summarized by the following theorem.

Theorem 5.1

The infection-free and predator-free steady state E 1 ( k , 0 , 0 ) of System (1) is stable if the parameters satisfy the condition k < μ λ .

(ii) At E 2 δ α 1 β , 0 , r ( k α 1 β − δ ) k α 1 2 β , the Jacobian matrix at E 2 is given by:

J E 2 = − r δ k α 1 β − δ ( r + k λ ) k α 1 β − δ β 0 ( δ λ − μ α 1 β ) α 1 β 0 0 0 2 r ( k α 1 β − δ ) − k α 1 δ k α 1 .

The eigenvalues associated with matrix J E 2 are λ 1 = − r δ k α 1 β < 0 , λ 2 = ( δ λ − μ α 1 β ) α 1 β , and λ 3 = 2 r ( k α 1 β − δ ) − k α 1 δ k α 1 . Using these eigenvalues, we have the following theorem.

Theorem 5.2

The infection-free steady state E 2 δ α 1 β , 0 , r ( k α 1 β − δ ) k α 1 2 β is stable if λ < μ α 1 β δ and 0 < r ( k α 1 β − δ ) < k α 1 δ 2 ; otherwise, it is unstable.

(iii) At E 3 μ λ , r ( k λ − μ ) λ ( r + k λ ) , 0 , the Jacobian matrix associated with E 3 is calculated as:

J E 3 = − μ r k λ − B μ k λ − α 1 μ 3 B 2 λ A r ( k λ − μ ) B 0 − α 1 r 3 ( k λ − μ ) 3 λ A B 0 0 − δ .

where A = [ ( μ ( r + k λ ) ) 2 + ( r ( μ − k λ ) ) 2 ] > 0 and B = ( r + k λ ) > 0 . The eigenvalues of matrix J E 3 are λ 1 = − δ < 0 , λ 2 = − ( A 2 B 2 r λ μ + A 4 B 4 r λ 2 μ ( r μ + 4 k λ ( μ − k λ ) ) ) 2 A 2 B 2 k λ 2 < 0 , and λ 3 = ( − A 2 B 2 r λ μ + A 4 B 4 r λ 2 μ ( r μ + 4 k λ ( μ − k λ ) ) ) 2 A 2 B 2 k λ 2 < 0 . These findings could be summarized by the following theorem:

Theorem 5.3

The predator-free steady state E 3 μ λ , r ( k λ − μ ) λ ( r + k λ ) , 0 is always stable if feasible.

(iv) At E * ( x * , y * , z * )

Since due to algebraic complexity of expression, the coexisting steady states E * ( x * , y * , z * ) of System (1) could not be obtained explicitly. Therefore, we discuss the stability of coexisting state E * ( x * , y * , z * ) by using the Routh-Hurwitz [31] criteria. The calculated Jacobian matrix at E * is as follows:

J E * = A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 ,

where A 11 = r ( k − 2 x * − y * ) k − α 1 x * 2 ( x * 2 + 3 y * 2 ) z * ( x * 2 + y * 2 ) − λ y * , A 12 = x * − r k + 2 α 1 x * 2 y * z * ( x * 2 + y * 2 ) 2 − λ , A 13 = − α 1 x * 3 ( x * 2 + y * 2 ) , A 21 = y * 2 α 2 x * y * 2 z * ( x * 2 + y * 2 ) + λ , A 22 = λ x * − μ − α 2 y * 2 ( 3 x * 2 + y * 2 ) z * ( x * 2 + y * 2 ) 2 , A 23 = − α 2 y * 3 ( x * 2 + y * 2 ) , A 31 = 2 β x * y * 2 z * 2 ( α 1 − α 2 ) ( x * 2 + y * 2 ) 2 , A 32 = 2 β x * 2 y * z * 2 ( α 2 − α 1 ) ( x * 2 + y * 2 ) 2 , and A 33 = 2 β z * ( α 1 x * 2 + α 2 y * 2 ) ( x * 2 + y * 2 ) − δ .

The characteristic equation associated with matrix J E * is given by:

λ 3 + P 1 λ 2 + P 2 λ + P 3 = 0 ,

where, P 1 = − T r ( J E * ) = − ( A 11 + A 22 + A 33 ) , P 2 = ( A 22 A 33 − A 23 A 32 ) + ( A 11 A 33 − A 13 A 31 ) + ( A 11 A 22 − A 12 A 21 ) , and P 3 = − D e t ( J E * ) . Thus, by using the Routh-Hurwitz criteria, we can conclude the following result.

Theorem 5.4

The coexisting steady state E * ( x * , y * , z * ) is locally asymptotically stable if all P i > 0 , i = 1 , 2 , 3 and P 1 P 2 > P 3 .

6 Bifurcation analysis

This section contains the qualitative behavior changes in the model System (1) due to the variation of some ecosystem parameters. We have observed the following qualitative changes (bifurcation) occurring in model System (1).

6.1 Transcritical bifurcation

The predator-free steady state E 3 μ λ , r ( k λ − μ ) λ ( r + k λ ) , 0 collides with infection-free and predator-free steady state E 1 ( k , 0 , 0 ) if μ = k λ . Due to this critical value of μ , the exchange of stability takes place in model System (1) around the steady state E 1 . This is happening due to the presence of transcritical bifurcation. It could be proved by using the Sotomayor’s theorem [23] in the following way.

If μ = μ T C = k λ , then Det ( J E 1 ) = 0 , and hence, the Jacobian matrix J E 1 has a zero eigenvalue. Let v and w be the eigenvectors of matrices J E 1 and J E 1 T corresponding to zero eigenvalues, respectively; then,

X = v 1 v 2 v 3 = − ( r + k λ ) r 1 0 , w = w 1 w 2 w 3 = 0 1 0 .

Furthermore, using Sotomayor’s theorem, we have

F μ ( E 1 ; μ T C ) = 0 0 0 , D F μ ( E 1 ; μ T C ) v = 0 0 0 0 − 1 0 0 0 0 v 1 v 2 v 3 = 0 − 1 0 .

Thus, Δ 1 = w T F μ ( E 1 ; μ T C ) = 0 , Δ 2 = w T [ D F μ ( E 1 ; μ T C ) v ] = − 1 ≠ 0 , and Δ 3 = w T [ D 2 F μ ( E 1 ; μ T C ) ( v , v ) ] = − 2 λ ( r + k λ ) r ≠ 0 . Hence, we can conclude the following theorem.

Theorem 6.1

System (1) experiences a transcritical bifurcation between E 1 and E 3 if the system parameters satisfy the condition μ = μ T C = k λ .

6.2 Hopf bifurcation

The characteristic equation of the Jacobian matrix corresponding to System (1) at E * ( x * , y * , z * ) is discussed by:

(3) λ 3 + P 1 λ 2 + P 2 λ + P 3 = 0 ,

where, P i ( i = 1 , 2 , 3 ) have the same meaning as discussed in Section (5). As it has been mentioned in Theorem (5.4), the stability of E * ( x * , y * , z * ) switches if the parameters of System (1) are varied and cross a critical value. Let us consider μ as the bifurcation parameter, and due to the variation in μ , stability switches as μ passes through a threshold value P 1 P 2 = P 3 . The conditions for occurring Hopf bifurcation are as follows:

( i ) P i > 0 , i = 1 , 2 , 3 , ( i i ) P 1 P 2 − P 3 = 0 , and ( i i i ) d d μ ( λ ( μ ) ) μ = μ H ≠ 0 , where, μ H is the root of P 1 P 2 − P 3 = 0 .

Differentiating equation (3) with respect to μ , we obtain

d λ d μ = − ( λ 2 d P 1 d μ + λ d P 2 d μ + d P 3 d μ ) ( 3 λ 2 + 2 P 1 λ + P 2 ) .

Let the purely imaginary eigenvalues be ± i ω at μ = μ H so that

d d μ ( Re ( λ ) ) λ = i ω = − d P 3 d μ − ω 2 d P 1 d μ ( P 2 − 3 ω 2 ) + 2 P 1 ω 2 d P 2 d μ ( P 2 − 3 ω 2 ) 2 + ( 2 ω P 1 ) 2 ,

where

d P 1 d μ = − 1 , d P 2 d μ = δ + λ y * + α 1 x * 2 z * ( x * 2 + 3 y * 2 ) ( x * 2 + y * 2 ) 2 − r ( k − 2 x * − y * ) k − 2 β z * ( α 1 x * 2 + α 2 y * 2 ) ( x * 2 + y * 2 ) , d P 3 d μ = − 1 k ( x * 2 + y * 2 ) 5 [ 2 k α 1 ( α 2 − α 1 ) β x * 4 y * 2 z * 2 ( x * 2 + y * 2 ) 2 − ( x * 2 + y * 2 ) 2 ( 2 z * ( α 1 x * 2 + α 2 y * 2 ) β − ( x * 2 + y * 2 ) δ ) ( r ( k − 2 x * − y * ) ( x * 2 + y * 2 ) 2 − k α 1 ( x * 2 + 3 y * 2 ) z * − k λ y * ( ( x * 2 + y * 2 ) 2 ) ) ] .

Now, we have to verify that:

d P 3 d μ − ω 2 d P 1 d μ ( P 2 − 3 ω 2 ) + 2 P 1 ω 2 d P 2 d μ μ = μ H ≠ 0 .

But it could be observed that the analytical verification of this expression is very complex. That is why, we verify it by using the following numerical example.

The chosen set of parametric values are as follows: r = 0.3 , k = 20 , α 1 = 0.0025 , α 2 = 0.05 , β = 0.15 , δ = 0.125 , and λ = 0.5 . The threshold value of the Hopf bifurcation parameter is μ H ≈ 0.0005 . Using these values the transversality condition of Hopf bifurcation could be verified.

Theorem 6.2

The coexisting steady state E * ( x * , y * , z * ) loses its stability through the Hopf bifurcation if the following conditions hold:

J E * has two purely eigenvalues and P i > 0 , i = 1 , 2 , 3 along with P 1 P 2 − P 3 = 0 ,
d d μ ( λ ( μ ) ) μ = μ H ≠ 0 , where, μ H is the root of P 1 P 2 − P 3 = 0 .

7 Numerical simulations

This section contains the numerical support of mathematical findings of System (1).

The chosen ecosystem parametric values for Figure 1 are as follows: r = 0.3 , k = 20 , α 1 = 0.0025 , α 2 = 0.05 , μ = 0.0005 , β = 0.15 , and δ = 0.125 . For Figure 1(a), the infectious contact rate is assumed to be λ = 0.2 and solution curve approaches to a stable coexisting state for this set of parametric values. But as small changes in infectious contact rate, i.e., λ = 0.2 to 0.5, occur, then System (1) oscillates around the positive steady state (refer to Figure 1(b)). Figure 1(c) and (d) shows the phase portraits corresponding to Figure 1(a) and (b), between susceptible and infected populations, respectively. Figure 1(e) and (f) depicts the phase portraits corresponding to Figure 1(b) between infected and predator and predator and susceptible populations, respectively.

Figure 1

(a) Solution curve for λ = 0.2. (b) Solution curve for λ = 0.5. (c) phase portrait corresponding to (a) in x–y axis. (d) phase portrait corresponding to (b) in x–y axis. (e) phase portrait corresponding to (a) in y–z axis. (f) phase portrait corresponding to (b) in x–z axis.

In Figure 2, the point B P is the point of bifurcation at which stability exchanges (transcritical bifurcation). The chosen parametric values for this plotting are as follows: r = 0.3 , k = 20 , λ = 0.05 , α 1 = 1.05 , α 2 = 0.65 , β = 0.25 , and δ = 0.02 .

$Figure 2 Transcritical bifurcation diagram around steady state E 1 {E}_{1} of System (1) with respect to μ \mu .$

Figure 2

Transcritical bifurcation diagram around steady state E 1 of System (1) with respect to μ .

8 Conclusion

Switching behavior in predation is a type of consumption technique of predator, which is frequency dependent, i.e., predator favors to consume that prey which are large in number. In this work, we consider a mathematical prey-predator model in which prey is suffering with micro-parasite infection along with predator switching strategy. Prey grows logistically, and the interaction terms of predation follow the switching function mechanism. The analysis reveals the facts that if the net growth of predator is not greater than its mortality rate, then in long run, predator population will not survive. In the presence of switching mechanism, four types of steady states are possible: infection and predator-free, infection-free, predator-free, and coexisting steady state. Among these, predator-free steady state is always stable and the stability of remaining states is conditionally dependent. With respect to the mortality rate of infected prey, the stability exchange between predator-free and infection and predator-free state, i.e., transcritical bifurcation, happens. The occurrence of Hopf bifurcation ensures the possibilities of survival of all species. Numerical simulations support the transcritical and Hopf bifurcation.

Funding information: This work was supported by Core Research Grant, Science Engineering Research Board, Government of India (CRG/2021/006380).
Conflict of interest: The authors declare that they have no conflict of interest.
Ethical approval: This research did not require any ethical approval.

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Received: 2023-04-21

Revised: 2023-08-12

Accepted: 2023-08-31

Published Online: 2023-11-03

This work is licensed under the Creative Commons Attribution 4.0 International License.

An eco-epidemiological model with predator switching behavior

Abstract

1 Introduction

2 Model formulation

3 Positivity and boundedness of solutions

Theorem 3.1

Proof

3.1 Extinction scenarios

Theorem 3.2

4 Steady states

5 Stability analysis

Theorem 5.1

Theorem 5.2

Theorem 5.3

Theorem 5.4

6 Bifurcation analysis

6.1 Transcritical bifurcation

Theorem 6.1

6.2 Hopf bifurcation

Theorem 6.2

7 Numerical simulations

8 Conclusion

References

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