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1. Introduction
In mathematical ecology, most of the researchers considered the prey-predator system in a homogeneous environment. But in a real life situation, the environment is heterogeneous, which contains different patches connected through migration. Many ecologists and researchers ([1–5]) have studied the impact of predator species migration on prey-predator interactions. In their work, the prey density was significant and the predator species was considered to remain in the specified patch. Many researchers comprehensively studied the dispersal model in a multipatch environment ([6–20]). Amarasekare [13] studied two-patch models of single species with local density-dependent dispersal and spatial heterogeneity. Padrón and Trevisan [18] considered a single species’ logistic growth model composing several habitats connected by linear migration. Stephen Cantrell et al. [19] examined evolutionary stability strategies for dispersal in heterogeneous patchy environments. In most cases, the randomness of the dispersal rate between different patches was assumed to be fixed. Kang et al. [20] formulated a Rosenzweig-MacArthur prey-predator model in two patch environments. The rate of dispersal has a great influence on stable prototypes and the persistence of the species shown in various research works ([21–24]). Ruxton [22] investigated the stability behaviour of a population model by adding density-dependent migration between nearest-neighbour populations. Rohani and Ruxton [23] presented a two-species interaction to establish the non-stabilization of the system for density-independent movement between populations. Fretwell and Lucas [25] proposed ideal free distribution, i.e. individuals in different patches possess similar fitness. Several researchers have already discussed this concept after that ([10, 26, 27]).
In balance dispersion [10], the dispersion rate remains constant when the population is in the equilibrium state, providing ideal free distribution. But constant rates of dispersal for the passive animal can lead the prey-predator system to a stable situation ([10, 28]). Furthermore, passive dispersal may create negative density dependence staffing rates of the population to stabilize the prey-predator system at an equilibrium state. Therefore, due to heterogeneity in patch or dispersal rates, an unstable population may behave like a stable population. Dispersal rates cannot be considered very high because they may harmonize local stability behaviour between patches. Therefore, to show the effect of dispersal, researchers have developed their models with the help of fixed dispersal rates. The population density alters very fast because of the dispersal process in most of their models. Krivan and Sirot [27] observed that, the stable population dynamics for two competitive species becomes unstable due to rapid adaptive animal dispersal. Abrams et al. ([29, 30]) also observed the profound impact of the dispersal process on the population dynamics.
Modeling the prey-predator system incorporating the dispersal concept is an active field of research. The dispersal rate of prey species steadily increases with the enlargement of predators in a patch ([31, 32]). Another exciting matter in this field is the Allee effect, a mechanism where the prey population shows a negative growth rate at low density ([33, 34]). There are several reasons to consider the Allee effect: mate restrictions, stumpy probability of victorious meeting [35], food mistreatment, supportive defence, inbreeding dejection [36], etc. Predator evasion [37] for evolutionary alterations is another important cause to consider the Allee effect. The strong Allee effect ([38–43]) has been considered here, whereas the concept of the weak Allee effect ([44, 45]) also exists in this regard. This classification depends on the per capita enlargement rate of the population at low density. The mathematical formulation of the growth equation due to the Allee effect takes the form
Recently, Pal and Samanta [61] presented the dispersal dynamics of the Allee effect in one prey of a prey-predator system in two patch environment. Saha and Samanta [62] studied the dispersal dynamics of a prey-predator system in two patch environments with two prey species considered the Allee effect. Consequently, we are introducing a prey-predator system with dispersal and strong Allee effect in three-patch environments, namely Patch 1, Patch 2 and Patch 3. Every patch consists of a pair of prey-predator species and a strong Allee effect in the prey population escalation in the first two patches. Our main objective is to study the impact of dispersal speed and the Allee effect on the population dynamics of our desired system. Ecology always demands a balance among its inhabitants, so stability of the population is always expected in this regard. We have studied the effect of dispersal and the Allee effect on the persistence of species. So, we hope that this study may help ecologists in their development of the environment.
The next section reflects the mathematical structure of the prey-predator interaction in three-patch environment. Section 3 is equipped with the positivity and boundedness characteristics for our proposed prey-predator system. Equilibrium points and their stability conditions have been discussed in Section 4. Bifurcation behaviour of the proposed model has been analysed in Section 5. Numerical verifications have been done in Section 6 and the concluding remarks have been given in Section 7.
2. Formation of Three-Patch Prey-Predator System
We opt for a six species (three each prey and predators) prey-predator system in three-patch
Notations:
Assumptions:
(i) Prey population growth rate is affected by the Allee effect in the first two patches.
(ii) All three predator species are free from the Allee effect.
(iii) Prey species are movable to higher fitness patches.
(iv) Conversion rate of prey biomass to predator biomass less than the predation rate
(v) We consider balanced dispersal ([11, 63]), mathematically
The graphical view of our proposed system of three-patch environment system is presented in Figure 1.
[figure(s) omitted; refer to PDF]
Based on the above notations, assumptions, and flow diagram, the population-dispersal dynamics can be presented by the following set of nonlinear differential equations:
Since we considered the balanced dispersal, and hence mathematically, we can write
When
3. Positivity and Boundedness of the Proposed System
Let us consider the following theorems to ensure that the anticipated model equation (1) is well-posed.
Theorem 1.
Each solution of the anticipated system equation (1) starting from
Proof.
The right-hand side of the model equation (1) is locally Lipschitzian in the space of continuous functions.
Therefore, the solution
Firstly we prove that
If it is false, then there exist
Now, the first equation of the system equation (1) provides,
Next, we claim that
Now, the second equation of the system equation (1) gives
We further claim that
Now, from the third equation of equation (1), we get
Again, from the last three equations of the system equation (1) we can obtain,
Theorem 2.
For balanced dispersal, every solution of the anticipated system equation (1) which starts from
Proof.
To prove this theorem, we have the following cases.
Case 1.
For
Now, for all
Now,
Therefore,
But for all
Thus,
Similarly, for
For
Now,
Then,
But
Therefore, we get
Case 2.
For
For
Now,
Then,
But
Therefore, we get
Case 3.
For
For
Now, for all
Now,
Therefore,
But for all
Thus,
Therefore, in this case we also get
Case 4.
For
Therefore, we get
Case 5.
For
Therefore, we get
Case 6.
For
Consequently, we get
Case 7.
For
Therefore, we get
Case 8.
Here we claim
Let us assume that
Now taking
Now,
Again, the first equation of (1) gives,
Also, we have,
This implies that
Similarly, we can prove that,
Lastly let us assume that
Now taking
Now,
Again, the third equation of (1) gives,
We also have,
Therefore, we conclude
Let
Then,
Now, if we take
Therefore,
This implies,
Now applying the theory of differential inequality, we get,
Taking
Then all the solutions of (1) that initiate in
This proves the theorem.
4. Equilibria and Stability of the Proposed Model
For the non-trivial equilibrium point of the proposed model (1) in the absence and presence of dispersal, we have
Case 9.
Clearly, the non-trivial equilibrium
From equations (37), and (39) we have
Thus,
Now, let us take that,
Also, let
Therefore, the condition (40) is verified and we get
Again from (36) we get,
Therefore,
Thus, we conclude
Case 10.
In this case, we have to consider the following set of equations to evaluate the interior equilibrium
The positive solution of equations (46)–(51) is the interior equilibrium
4.1. Stability Condition of the Proposed System without Dispersal
At this point
Theorem 3.
The system (1) is unstable without dispersal
Proof.
The characteristic equation of the variation matrix
Evaluating, we get (see Appendix A),
Therefore, we have,
Now, we observe that
Again
Then all the roots of (57) and (58) have positive real parts. Thus, the system (1) is unstable without dispersal
4.2. Persistence of the Proposed System in the Presence of Dispersal
The biological meaning of persistence is the survival of all populations in future time. For the Kolgomorov type equations, persistence was discussed by Freedman and Waltman [64]. Mathematical definition for persistence of a system can be written as follows:
Definition 1.
(Persistence): if
Theorem 4.
The system (1) persists.
Proof.
Consider the matrix:
Here
Here we can recall the theorem which states that,
Which implies, (see Appendix B)
4.3. Stability in the Presence of Dispersal
Let us consider the understated theorem for the stability of the equilibrium
Theorem 5.
The equilibrium
Proof.
The characteristic equation for
Taking
And expanding the determinant we obtain,
Simplifying we get,
Now, the Hurwitz matrix
The roots of the auxiliary equation have negative real parts if and only if all the principal diagonal minors of the Hurwitz matrix are positive, provided that
5. Hopf Bifurcation
When the parameter of a dynamical system changes due to a sudden qualitative change in its behavior, then a bifurcation of the system occurs. Hopf bifurcation occurs in the case of nonhyperbolic nonlinear equations for two or more dimensions. It is typically happened in a differential equation for switching the eigenvalues to become purely imaginary. The fixed point switches from a stable focus to an unstable one when the real part of the eigenvalue changes from negative to positive; such a bifurcation is known as supercritical. On the contrary, the bifurcation is subcritical when the fixed point changes from an unstable focus to a stable one for switching the real part of the eigenvalue from positive to negative.
We observed a special type of Hopf bifurcation for the system proposed in this study. We find a pair of complex conjugate eigenvalues that pass through the imaginary axis of the Jacobian matrix, and all other eigenvalues consist of negative real parts. Bifurcated limit cycles can be observed in a supercritical Hopf bifurcation both physically and numerically. To distinguish this type of bifurcation with the same for the eigenvalues on the right half-plane, let us call them a simple Hopf bifurcation. In this study, we are interested in exploring the possible occurrence of a simple Hopf bifurcation around
The following theorem provides the necessary and sufficient condition for the occurrence of a simple Hopf bifurcation of the system (1):
Theorem 6.
The necessary and sufficient conditions for the system (1) to undergo Hopf bifurcation at
Proof.
The characteristic equation for
The Hurwitz matrix
Using the condition for simple Hopf bifurcation [66] at
6. Numerical Simulation
For the numerical verification of analytical findings of the previous sections, computer simulations have been performed to obtain a different graphical presentation of the proposed model. We used Matlab and Wolfram Mathematica for this purpose, and the figures obtained are extremely imperative from a realistic viewpoint. We have used a set of hypothetical data to examine the effect of dispersal speed on the patches. So, we decided to do the simulations in two different cases. Case 11 attributes to the prey-predator system when the prey population budges independently in its patches, i.e., dispersal does not happen
Table 1
Numerical values of the model parameters.
Parameter | Value | Parameter | Value |
Case 11.
Simulations without dispersal
Based on the model parameters as prearranged in Table 1, we achieve the inner equilibrium point
Clearly, the system becomes unstable with time progress in the absence of dispersal (Figure 2). Without dispersal speed
From Figures 3 to 5, we observe that with the increasing values of
Now, the impact of the mortality rate of the predator species
From Figures 7 and 8 we observed that, as
We have studied the influence of environmental carrying capacity and death rate of predators on species without dispersal. We found that parameters such as
[figure(s) omitted; refer to PDF]
Case 12.
Simulations with dispersal
Here we assume the positive dispersal of the system (1) and taking
The time series graph of the system (1) for
Figure 11 reflects the stable behavior of the proposed prey-predator system in the presence of dispersal. Here we found that each of the species of system (1) gradually converges to the inner equilibrium point
The effect of
From Figure 12, we observe that the predator species of the first patch gradually increases and all other species remain constant with the progression of
Again, to observe the effect of predator mortality rate on species, we consider Figures 16 to 19.
From Figure 16, we observe that, the predator species in patch one gradually decreases. Consequently, prey species in patch one gradually increases. Also, predator species in patches two and three increase progressively, and the remaining two prey species maintain constant levels. The same types of observation are seen at in Figures 17 and 18, respectively. In Figure 19, we consider
Finally, we observe the effect of dispersal speed
Table 2 reflects that the population level of the prey species
The pictorial presentation of the trophic cascade of the system (1) with increasing dispersal speed is described in Figure 20.
Here we found that, the predator population is very sensitive to changing values of
Now, if we gradually increase the value of
All figures show a subcritical Hopf bifurcation localized at
[figure(s) omitted; refer to PDF]
Table 2
Equilibrium points for different
Equilibrium point ( | |
[figure(s) omitted; refer to PDF]
7. Discussion and Conclusion
In this study, a three-patch based prey-predator system is presented with a strong Allee effect in the first two patches. This is the first attempt to study the dynamics of the prey-predator system in three-patch environments with the Allee effect in the first two patches. The impact of the dispersal speed on the stability and persistence of the proposed system is elaborately studied in this paper. For a better understanding of the proposed model, we provide the concept of balanced dispersal. We observed that the equilibrium level of the prey species depends on the mortality rate of the predator species as well as the conversion rate of the prey biomass to the predator biomass. These population levels are not affected by the speed of dispersal. But the speed of dispersal has a definite effect on the equilibrium level of predator species. In addition, the stable nature of the equilibrium point depends on the mortality rate of the predator in each patch. The stability criterion of the system in the absence of dispersal (Theorem 3) can help to maintain the ecological balance. But the system becomes stable in the presence of positive dispersal speed (Theorem 5). Therefore, we conclude that the mortality rate of the predator species and the dispersal speed play a vital role in the dynamics of the supposed ecological system.
The numerical verification of the analytical findings is vital to the biological relevance of this type of model. Moreover, graphical views always support the findings and contribute to the vast future development of the ecosystem. Therefore, the proposed system may contribute to the dynamics of the prey-predator in the assumed situations.
Finally, we conclude that the proposed system in three-patch environments is very interesting and shows complex dynamics. But the model can also be modified by taking the Allee effect in all three patches. Predator species can also be assumed to move between patches, making the model more interesting and realistic.
A. Proof of Theorem 3
The characteristic equation for
Expanding The authors get,
Therefore,
B. Proof of Theorem 4
The characteristic equation for the matrix
Taking
Expanding The authors evaluate,
Simplifying The authors get,
Therefore,
C. Proof of Theorem 5
The authors have the characteristic equation,
Simplifying The authors obtain.
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Abstract
This paper presents the dispersal effect on a prey-predator model with three-patch states incorporating a strong Allee effect on the first two prey populations. The prey species are considered to be mutable to exhibit a balanced dispersal between the patches. The dispersal among patches is directed through lower fit patches to higher fit patches. This paper derives a new approach for dispersal and Allee’s effect with the specific condition on the stability on a three-patch of a three-species prey-predator anticipated system. The persistence of the system is observed because of the dispersal effect on the three-patch prey-predator system. The numerical simulations of analytical findings are presented using hypothetical parameter values to relate real-world prey-predator situations for balancing ecology.
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1 Department of Mathematics, National Institute of Technology Puducherry, Karaikal 609609, India
2 Indian Institute of Engineering and Science and Technology, Shibpur, West Bengal 711103, India
3 Maulana Abul Kalam Azad University of Technology, Kolkata 700064, West Bengal, India
4 Department of Mathematics Education, Akenten Appiah-Menka University of Skills Training and Entrepreneurial Development, Kumasi, Ghana