Analysis of an Eco-Epidemiological Model Considering Effect of Harvesting and Prey Refuge

In this work, we analyze an eco-epidemiological model with the disease in the prey, considering a constant proportion of harvesting of either species or a prey refuge. The positive invariant set, the conditions of existence, and locally asymptotically stability of the equilibria are studied using the stability theory of ordinary differential equation. The global stability of border equilibria by constructing Lyapunov functions and permanence of the system by comparison theoremare proved. The numerical simulation further proved the correctness of the theoretical analysis. The result indicates that overfishing would lead to population extinction and a reasonable fishing strategy should keep the coexistence of populations.

• Refuge
• Harvesting
• Eco-Epidemiological Model
• Permanence

1. Introduction

Eco-epidemiology is a new offshoot of biological mathematics which considers the dynamics of infectious disease spreading among the ecosystem, which has already been researched in a lot of articles [1] [2]. They discussed the predator-prey system with the disease among prey population [3], predator population [4], both predators and prey populations.

The effect of refuges is vital on the ecosystem population, and research on the dynamic behaviors of incorporating prey refuges has become the main topic during the last decade [5] [6] [7]. A prey refuge is taken into account from two types. One type is a fixed proportion of prey refuge: ( 1 − ν ) x , where ν ∈ ( 0,1 ) , another type is a fixed quantity of prey refuge: x − ν . A prey refuge may lead to population extinction or system’s fluctuations. For example, a destabilization phenomenon appears because of prey refuge in an eco-epidemiological system [8]. Choosing different prey refuge decides whether undergoes Hopf bifurcation [9].

Reasonable harvesting policy also has been done for a long time of research, which is one of the interesting and main problems in economics and ecology [10]. For example, a predator-prey model with constant proportion of prey harvesting is studied [11]. Biwen Li et al. [12] researched a predator-prey model taking into account nonselective harvesting. The complex behavior of an eco-epidemiological model considering prey catching was investigated by S.N. Raw and Barkha Tiwari [13].

Based on the previous research, we formulate an eco-epidemiological model, considering a constant proportion of harvesting of either species or a prey refuge. The first purpose of the paper is to study the complex behavior (stability, branches and permanence) of the model. The second purpose is to study the impact of harvesting of different populations on the extinction of disease.

The organization of the paper is as follows: In Section 2, a prey-predator model with the disease in the prey, considering a constant proportion of harvesting of either species or prey refuge is built. The conditions of stability of equilibria and the sufficient condition for the permanence of the model are obtained in Section 3. Furthermore, in Section 4, we analyze the impact of harvesting of different populations. The correctness of the theoretical analysis is verified by numerical simulation in Section 5. And our findings are briefly discussed in Section 6.

2. The Mathematical Model

Assumptions: 1) In the existence of infection, prey population is separated into two parts—the susceptible prey S ( t ) , the infected prey I ( t ) . The whole prey population grows in terms of logistic growth with carrying capacity k and an intrinsic birth rate r, namely

N ˙ ( t ) = r N [ 1 − N k ] ,     where   N ( t ) = S ( t ) + I ( t ) .

2) The disease is only spreading among prey populations and not inherited. The infected population doesn't recover. Because of infection, infected preys are caught easily. The predator only consumes infected prey according to the Holling III functional response. A conversion rate of consumed prey is η .

3) Only the susceptible preys have reproductive capacity. The death rates of predator and infected prey are d, c, respectively.

4) m ( 0 < m < 1 ) is a fixed rate of infected prey using refuges.

5) E 1 > 0 , E 2 > 0 , E 3 > 0 express the harvesting capabilities for the predator, susceptible prey, the infected prey, respectively. q 1 E 1 S , q 2 E 2 I , and q 3 E 3 Y , represent the catch of the respective species, where q 1 , q 2 , q 3 represent the catch efforts coefficients of the predator, the susceptible prey and the infected prey respectively. All the parameters mentioned are supposed to be positive.

We construct the following model.

{ S ˙ ( t ) = r S ( 1 − S + I k ) − β S I − q 1 E 1 S I ˙ ( t ) = β S I − ( 1 − m ) 2 I 2 Y b + ( 1 − m ) 2 I 2 − q 2 E 2 I − c I Y ˙ ( t ) = − d Y + η ( 1 − m ) 2 I 2 Y b + ( 1 − m ) 2 I 2 − q 3 E 3 Y (2.1)

Theorem 2.1 All solutions of system (2.1) satisfying initial conditions in R³ are ultimately bounded.

Proof:Let ϖ = η S + η I + Y , then

d ϖ d t = η d S d t + η d I d t + d Y d t = r S [ 1 − S k ] − η r k S I − η q 1 E 1 S − ( η c + η q 2 E 2 ) I − ( d + q 3 E 3 ) Y ≤ η r S − η q 1 E 1 S − ( η c + η q 2 E 2 ) I − ( d + q 3 E 3 ) Y ≤ η r k − [ η q 1 E 1 S + ( η c + η q 2 E 2 ) I + ( d + q 3 E 3 ) Y ] ≤ η r k − θ ϖ

where 0 < θ < min { η q 1 E 1 , ( η c + η q 2 E 2 ) , ( d + q 3 E 3 ) } then

d ϖ d t + θ ϖ ≤ μ r k

The linear differential inequality is solved to obtain the result.

0 < ϖ ( S , I , Y ) < η r k θ ( 1 − e − θ t ) + ϖ ( S ( 0 ) , I ( 0 ) , Y ( 0 ) ) e − θ t

If t → ∞ , 0 < ϖ < η r k θ

The model (2.1) of all solutions starting in ( S ( 0 ) , I ( 0 ) , Y ( 0 ) ) ∈ R + 3 remain in R 3 for all t ≥ 0 . Thus, all trajectories of system (2.1) will enter the area: M = { ( S , I , Y ) ∈ R + 3 : ϖ = η r k θ + ζ , ζ > 0 }

3. Stability Analysis

3.1. Existing of Equilibrium Points

The system (2.1) has several equilibria:

1) The zero equilibrium E a = ( 0,0,0 ) ,

2) The axial equilibrium E b = ( k ( r − q 1 E 1 ) r ,0,0 ) ,

3) The boundary equilibrium E c = ( S 2 , I 2 ,0 ) , and S 2 = c + q 2 E 2 β , I 2 = r k β − r c − r q 2 E 2 − k β q 1 E 1 β ( r + k β ) ,

4) The positive equilibrium E ∗ = ( S ∗ , I ∗ , P ∗ ) where S ∗ = r k − ( r + k β ) I ∗ − k q 1 E 1 r , I ∗ = b ( d + q 3 E 3 ) ( η − d − q 3 E 3 ) ( 1 − m ) 2 , Y ∗ = ( β s ∗ − c − q 2 E 2 ) ( b + ( 1 − m ) 2 I ∗ 2 ) ( 1 − m ) 2 I ∗ 2 ,

If r > q 1 E 1 , the equilibrium point E b exists;

If r k β > r c + r q 2 E 2 + k β q 1 E 1 , the equilibrium points E c exists;

If η − d − q 3 E 3 > 0 , r k − ( r + k β ) I ∗ − k q 1 E 1 > 0 , and β s ∗ − c − q 2 E 2 > 0 , the equilibrium point E ∗ exists.

3.2. Locally Stability

Let E ¯ = ( S ¯ , I ¯ , Y ¯ ) is an arbitrary equilibrium point of the model (2.1), the Jacobian matrix at arbitrary equilibrium point is

| r − 2 r k S ¯ − r k I ¯ − β I ¯ − q 1 E 1 − r k S ¯ − β S ¯ 0 β I ¯ β S ¯ − 2 b ( 1 − m ) 2 I ¯   Y ¯ [ b + ( 1 − m ) 2 I 2 ] 2 − ( c + q 2 E 2 ) − ( 1 − m ) 2 I 2 b + ( 1 − m ) 2 I ¯ 2 0 2 η b ( 1 − m ) 2 I ¯   Y ¯ [ b + ( 1 − m ) 2 I ¯ 2 ] 2 − d + η ( 1 − m ) 2 I ¯ 2 b + ( 1 − m ) 2 I ¯ 2 − q 3 E 3 | (3.1)

By calculating the positive or negative of the real part of the eigenvalue, the local asymptotic stability of each equilibrium point can be judged. The details are as follows:

1) λ 1 = r − q 1 E 1 , λ 2 = − ( c + q 2 E 2 ) , λ 3 = − ( d + q 3 E 3 ) are three eigenvalues of the Jacobian matrix at E a . The conditions of local asymptotical stability of E a are r < q 1 E 1 , λ 1 < 0

2) λ 1 = − ( r − q 1 E 1 ) , λ 2 = r k β − r c − r q 2 E 2 − k β q 1 E 1 r , λ 3 = − ( d + q 3 E 3 ) are three eigenvalues of the Jacobian matrix at E b . If r k β < r c + r q 2 E 2 + k β q 1 E 1 , λ 2 < 0 , E b is locally asymptotically stable

3) Concerning E c , its characteristic equation is

μ 3 + A 1 μ 2 + A 2 μ + A 3 = 0

where A 1 = − ( a 11 + a 33 ) , A 2 = a 11 a 33 − a 12 a 21 , A 3 = a 12 a 21 a 33 and a 11 = − r S 2 k , a 12 = − ( r k − β ) S 2 , a 21 = β I 2 , a 33 = − d + η ( 1 − m ) 2 I 2 2 b + ( 1 − m ) 2 I 2 2 − q 3 E 3 .

If η − d − q 3 E 3 < 0 , a 33 < 0 , And A 1 > 0 , A 3 > 0 , A 1 A 2 − A 3 > 0 . According to the Routh-Hurwitz rule, all eigenvalues of the Jacobian matrix at E c are negative. The local asymptotical stability of E c is obtained.

4) Concerning E ∗ , its characteristic equation is

μ 3 + B 1 μ 2 + B 2 μ + B 3 = 0

where B 1 = 2 r k S ∗ + β S ∗ ,

B 2 = − r k S ∗ [ ( 1 − m ) 2 I ∗ 2 Y ∗ b + ( 1 − m ) 2 I ∗ 2 − 2 b ( 1 − m ) 2 I ∗ Y ∗ ( b + ( 1 − m ) 2 I ∗ 2 ) 2 ]     + β ( r k + β ) S ∗ I ∗ + 2 b η ( 1 − m ) 4 I ∗ 3 Y ∗ ( b + ( 1 − m ) 2 I ∗ 2 ) 3

B 3 = 2 a r η ( 1 − m ) 4 S ∗ I ∗ 3 Y ∗ k ( b + ( 1 − m ) 2 I ∗ 2 ) 3 .

Obviously, H 1 = B 1 > 0 , B 3 > 0 .

If η 2 b ( d + q 3 E 3 ) < 4 ( η − d − q 3 E 3 ) 3 ( 1 − m ) 2 , H 2 = B 1 B 2 − B 3 > 0 , so H 3 = B 3 H 2 > 0 .

In according with the Routh-Hurwitz rule, all eigenvalues of the Jacobian matrix at E ∗ are negative. The local asymptotical stability of E ∗ is obtained.

3.3. Globally Asymptotically Stable

The global asymptotically stability of each equilibrium point is judged by constructing the Lyapunov function.

Theorem 3.1 The condition of global asymptotical stability of E a is r k − r − k q 1 E 1 < 0 on plane Γ = { ( S , I , Y ) : S > 0 , I > 0 , Y > 0 } .

Proof: The Lyapunov function V 1 = S + I + Y μ is constructed, then we have

d V 1 d t = r ( 1 − S k ) S − q 1 E 1 S − r k S I − ( c + q 2 E 2 ) I − 1 θ ( d + q 3 E 3 ) Y ≤ ( r − r k − q 1 E 1 ) S − ( c + q 2 E 2 ) I − 1 θ ( d + q 3 E 3 ) Y

If r k − r − k q 1 E 1 < 0 , d V d t ≤ 0 And,

D 1 = { ( S , I , Y ) ∈ M | d V 1 d t = 0 } = { ( S , I , Y ) = ( 0 , 0 , 0 ) } .

So, the global asymptotical stability of E a is proved.

Theorem 3.2 The condition of global asymptotical stability of E b is β k − c − q 2 E 2 < 0 on plane Γ = { ( S , I , Y ) : S > 0 , I > 0 , Y > 0 } .

Proof: A Lyapunov function V 2 = η I + Y is constructed, then we obtain that

d V d t = η ( β S − c − q 2 E 2 ) I − ( d + q 3 E 3 ) Y ≤ η ( β k − c − q 2 E 2 ) I − ( d + q 3 E 3 ) Y

Hence, if β k − c − q 2 E 2 < 0 , d V d t ≤ 0 .

and

D 2 = { ( S , I , Y ) ∈ M | d V d t = 0 } = { ( S , I , Y ) = ( S , 0 , 0 ) }

the maximum invariable set of system (2.1) is Γ 1 = { I = 0 , Y = 0 } .

So the limit equation of system (2.1) is

d S d t = r S ( 1 − S k ) − q 1 E 1 S

Obviously, the global asymptotic stability of E b is obtained.

Theorem 3.3. The condition of global asymptotical stability of E c is d + q 3 E 3 − η > 0 on plane Γ = { ( S , I , Y ) : S > 0 , I > 0 , Y > 0 }

Similarly, The Lyapunov function V 3 = Y is constructed,

d V 3 d t = − [ d − η ( 1 − m ) 2 I 2 b + ( 1 − m ) 2 I 2 + q 3 E 3 ] Y = − ( d + q 3 E 3 ) b + ( d + q 3 E 3 − η ) ( 1 − m ) 2 I 2 b + ( 1 − m ) 2 I 2 Y

If d + q 3 E 3 − η > 0 , obviously d V d t ≤ 0 , and

D 3 = { ( S , I , Y ) ∈ B | d V d t = 0 } = { ( S , I , Y ) = ( S , I , 0 ) }

the maximum invariable set of system (2.1) is Σ = { Y = 0 } . So, the global asymptotical stability of E c is obtained.

3.4. Branching Phenomenon

Theorem 3.4. If the following conditions are true, Hopf bifurcation occurs at the equilibrium point of E ∗ .

i) If E 2 = E 2 ∗ , B 1 > 0 , B 3 > 0 ;

ii) If E = E 2 ∗ , B 1 B 2 − B 3 = 0 , the eigenvalue of E ∗ has a pair of double virtual roots;

iii) R e [ d μ j d E 2 ] E 2 = E 2 ∗ ≠ 0 , ( j = 1 , 2 , 3 ).

Proof:

According to the previous characteristic equation of E ∗ , if B 1 B 2 − B 3 = 0 , obtaining θ = θ ∗ , the characteristic equation of E ∗ can be written as

( μ 2 + B 2 ) ( μ + B 1 ) = 0

The three eigenvalues are i B 2 , − i B 2 , − B 1 .

Simultaneous derivation of E 2 on both sides of the equation, obtaining

3 μ 2 d μ d E 2 + d B 1 d E 2 μ 2 + 2 B 1 μ d μ d E 2 + d B 2 d E 2 μ + B 2 d μ d E 2 + d B 3 d E 2 = 0

Namely,

d μ d E 2 E 2 = E 2 ∗ = − [ d Q 1 d E 2 μ 2 + d Q 2 d E 2 μ + d B 3 d E 2 3 μ 2 + 2 B 1 μ + Q 2 ] μ = i B 2 = − [ d ( B 1 B 2 − B 3 ) d μ 2 ( B 1 2 + B 2 ) ] E 2 = E 2 ∗ + i [ B 2 d B 2 d E 2 2 B 2 − B 1 d ( B 1 B 2 − B 3 ) d E 2 2 B 2 ( B 1 2 + B 2 ) ] E 2 = E 2 ∗

In other words, R e [ d μ d E 2 ] E 2 = E 2 ∗ = − [ d ( B 1 B 2 − B 3 ) d μ 2 ( B 1 2 + B 2 ) ] E 2 = E 2 ∗ ≠ 0

3.5. Permanence

Theorem 3.5. If r k β < r c + r q 2 E 2 + k β q 1 E 1 and ( r k β − r c − r q 2 E 2 − k β q 1 E 1 ) 2 ( η − d + q 3 E 3 ) > [ β ( r + k β ) ] 2 ( d + q 3 E 3 ) b , the system (2.1) is permanence.

Proof: For the sake of proving the permanence of system (2.1), a average Lyapunov function V ( S , I , Y ) = S α 1 I α 2 Y α 3 is constructed, where α 1 , α 2 , α 3 is positive in the plan R 3 ,

V ′ V = α 1 ( r ( 1 − S + I k ) − β I − q 1 E 1 ) + α 2 ( β S − c − ( 1 − m ) 2 I Y b + ( 1 − m ) 2 I 2 − q 2 E 2 )

α 3 ( − d + η ( 1 − m ) 2 I 2 a + ( 1 − m ) 2 I 2 − q 3 E 3 ) ≡ Φ ( S , I , Y )

The following conclusions need to be verified: Φ ( S , I , Y ) > 0 for each boundary equilibrium point.

Let α 1 > α 2 ( c + q 2 E 2 ) + α 3 ( d + q 3 E 3 ) , then

Φ ( E 0 ) = α 1 r − α 2 ( c + q 2 E 2 ) − α 3 ( d + q 3 E 3 )

Φ ( E 1 ) = α 2 [ β k ( r − q 1 E 1 ) r − q 2 E 2 ] − α 3 ( d + q 3 E 3 )

Φ ( E 2 ) = α 3 ( − d + μ ( 1 − m ) 2 I 2 2 b + ( 1 − m ) 2 I 2 2 − q 3 E 3 )

Obviously, Φ ( E a ) > 0 . If r k β < r c + r q 2 E 2 + k β q 1 E 1 , Φ ( E b ) > 0 .

If ( r k β − r c − r q 2 E 2 − k β q 1 E 1 ) 2 ( μ − d + q 3 E 3 ) > [ β ( r + k β ) ] 2 ( d + q 3 E 3 ) b , Φ ( E c ) > 0 .

4. The Impact of Harvesting

Three aspects are mainly discussed.

1) Taking into account the harvesting of susceptible prey E 1 .

∂ S ∗ ∂ E 1 = − k q 1 r < 0 , ∂ I ∗ ∂ E 1 = 0 , ∂ Y ∗ ∂ E 1 = 0

The increased the harvesting effort E 1 can decrease the quantity of susceptible prey and not change the quantity of infected prey and predator. If E 1 < r k − r k q 1 , the three population coexist. In other words, the disease exists. However, if E 1 > r q 1 , the prey and the predator all die out, which is the sufficiency condition of the globally asymptotically stable of E a . If r k − r k q 1 < E 1 < r q 1 , the equilibrium point E b exists, both infected prey and predator become extinct.

2) Taking into account the harvesting of infected prey E 2 .

∂ S ∗ ∂ E 2 = 0 , ∂ I ∗ ∂ E 2 = 0 , ∂ Y ∗ ∂ E 2 = − q 2 ( 1 − m ) 2 I 2 < 0

Increasing of E 2 can decrease the quantity of predator population and not change the density of infected prey and uninfected prey. However, if E 2 > k β − c q 2 , the predator dies out, which is the condition of the globally asymptotically stable of E b , which is so obvious, because the increasing of E 2 may result in the lack of food for predator.

3) The case is only considering the harvesting predator species.

∂ S ∗ ∂ E 3 = − ( r + k β ) b q 3 η ( 1 − m ) 2 2 r ( η − d − q 3 E 3 ) 2 ( 1 − m ) 2 b ( d + q 3 E 3 ) ( η − d − q 3 E 3 ) ( 1 − m ) 2     × b q 3 η ( 1 − m ) 2 ( η − d − q 3 E 3 ) 2 ( 1 − m ) 2 < 0

∂ I ∗ ∂ E 3 = b q 3 η ( 1 − m ) 2 2 ( η − d − q 3 E 3 ) 2 ( 1 − m ) 2 b ( d + q 3 E 3 ) ( η − d − q 3 E 3 ) ( 1 − m ) 2 > 0 ,

∂ Y ∗ ∂ E 2 = − q 2 ( 1 − m ) 2 I 2 = β [ b + ( 1 − m ) 2 I ∗ 2 ] ( 1 − m ) 2 I ∗ 2 ∂ S ∗ ∂ E 3 − 2 b ( 1 − m ) 2 I ∗ ( β S ∗ − c − q 3 E 3 ) ( 1 − m ) 2 I ∗ 4 ∂ I ∗ ∂ E 3 < 0

The increasing the harvesting effort E 3 can result in raising the quantity of infected prey and reducing the quantity of predator and susceptible prey. The condition of globally asymptotically stability of E c is E 3 > η − d q 3 . In other words, the predator becomes extinct.

5. Number Simulations

Numerical simulation is used to further prove the correctness of theoretical analysis.

Let r = 0.9 , k = 48 , β = 0.03 , c = 0.15 , m = 0.45 , b = 0.4 , d = 0.04 , η = 0.036 . By assuming some parameters from the perspective of practical problems, we may observe the system (1.2) has the various results:

By changing different parametric values of q₁, E₁, q₂, E₂, q₃, E₃, we may observe the system (1.2) has the various results:

Along with q₁, E₁ increasing, the infected preys I and the predators Y are beginning to tend to 0, see Figure 1(a). Along with q₃, E₃ increasing, the predators Y are beginning to tend to 0, see Figure 1(b). But, along with q₂, E₂ changing, the stability of the system (2.1) undergoes changes: from state (see Figure 1(c))

to unstable stable state see Figure 2(a) & Figure 2(b). In other words, the three populations S, I and Y all exist, see Figure 1(c). We can choose different harvesting of either species to control their existence.

Form numerical simulation, the refuge of prey don't influence the system'stability of, fulfilling the conditions of existence and stability of all equilibrium points.

6. Conclusions

This paper discusses an eco-epidemiological model with the disease among the prey, taking into account the prey refuge and a fixed rate of harvesting of every species. All results indicate that harvesting has an important effect on the eco-epidemiological system. Over-exploitation ( E 1 > r q 1 ) would result in the extinction of all populations (susceptible, infected prey and predator). A reasonable harvesting strategy ( r k − r k q 1 < E 1 < r q 1 ) should ensure the existence of the susceptible prey, whereas the infected prey and predator eventually tend to becomeextinct. And the increasing harvesting E₂ can reduce the quantity of predators and doesn’t change the density of infected prey and susceptible prey. However, if E 2 > k β − c q 2 , the predator dies out, because the increase of E₂ may result in a lack of food for predator. The increased harvesting effort E₃ can result in increasing the quantity of infected prey and a decrease in predator and susceptible prey. The susceptible prey, infected prey, and predator all coexist. If E 3 > μ − d q 3 , the predator becomes extinct. Therefore, it is probable to control the dynamic behavior of the model by choosing reasonable variables E₁, E₂ or E₃.

In the paper, we consider that a fixed rate of harvesting of every species is continuous. In fact, the harvesting of every species is discontinuous. Further studies are required to analyze the dynamics of more realistic but complex systems such as considering the effect of impulsive capturing in different species.

Founding

This work is supported by the National Sciences Foundation of China (11971278), the Science Foundation of Shanxi Province (202103021223395) and supported by Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (20210602).

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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