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.
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 where, represents the population size at time , is the per capita growth rate, environmental carrying capacity is presented by and is the Allee edge. Here, . The population change rate continuously diminishes ([46, 47]) and finally departs to extermination ([42, 43]) if the relation holds. On the contrary, at low population density, the population intensification rate decreases but stays positive in the weak Allee effect. Researchers ([48–51]) observed the impact of the Allee effect on the dynamical behaviour of the prey-predator system. Celik and Duman [52] enormously studied the stability behaviour of a prey-predator model with and devoid of the Allee effect. The impact of the Allee effect on a prey-predator harvesting system was investigated by Javidi and Nyamorady [53]. Wang et al. [54] discussed the prey-predator system that exhibits intricate population dynamics due to the Allee effect. Zhou et al. [55] studied the population dynamics of a prey-predator system with the Allee effect. Kent et al. [56] used the concept of Allee effect as prey outfluxes. Therefore, numerous works ([57–60]) on prey-predator interactions have already been presented in this field of ecological modeling. A small number of researchers studied the dispersal dynamics of the prey-predator system in a patchy environment with a strong Allee effect.
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 environments based on the following notations and assumptions.
Notations: : prey population density in patch . : predator population density in patch . : growth rate of prey species in patch . : environmental carrying capacity of the prey in patch in nonattendance of predation and dispersal. : threshold value of the Allee effect in patch without predation as well as dispersal . : predation rate in patch . : dispersal speed among patches. : probability of dispersion from the patch to the patch . : conversion rate of prey biomass to predator biomass in patch . : mortality rate of the predator species in patch .
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 , , i.e, there is no mesh progress among the patches.
The graphical view of our proposed system of three-patch environment system is presented in Figure 1.
Based on the above notations, assumptions, and flow diagram, the population-dispersal dynamics can be presented by the following set of nonlinear differential equations:with primary information and .
Since we considered the balanced dispersal, and hence mathematically, we can write , , . Therefore, the carrying capacities, which are identical to the population abundances in all patches, always imply no mesh progress among the patches. Consequently, this situation is precisely the same as without dispersal. Therefore, without dispersal speed, the proposed model equation (1) converts to the following system of equations:with initial densities and for all .
When , the population in each patch progresses separately. In that case, the non-trivial equilibrium can be found by solving the system of equations . Our primary target is to find the interior equilibrium point of the proposed system equation (1) with respect to , , , , , and we shall also observe the impact of the dispersal rate on the said equilibrium point.
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 stay positive for all time.
Proof. The right-hand side of the model equation (1) is locally Lipschitzian in the space of continuous functions.
Therefore, the solution along with the positive initial solution exists and is unique in the interval where .
Firstly we prove that for all .
If it is false, then there exist so that , as well as for all .
Now, the first equation of the system equation (1) provides,which contradicts and hence for all .
Next, we claim that for all . If it is false, then there exist so that , as well as for all .
Now, the second equation of the system equation (1) giveswhich again contradicts and consequently, for all .
We further claim that for all . If our claim is false, then their exist so that , as well as for all .
Now, from the third equation of equation (1), we getthis also contradicts . Hence, for all .
Again, from the last three equations of the system equation (1) we can obtain,therefore, from the above discussion, we can conclude as well as for all . Hence, the theorem is proved.
Theorem 2. For balanced dispersal, every solution of the anticipated system equation (1) which starts from is uniformly bounded there.
Proof. To prove this theorem, we have the following cases.
Case 1. , , .
For , let us claim that . Assume that the claim is false. Therefore, there exist and in such a way that as well as for all where and for all as , are the respective carrying capacities of patch 2 and patch 3.
Now, for all we have,where,Now,Therefore,But for all , as , and , .
Thus, which is a contradiction and proves that the claim is valid. Hence .
Similarly, for we can prove that taking and .
For , we claim . Again, if we assume that the claim is not true, then there exist , such that and for all where and as , are the respective carrying capacities of patch 1 and patch 2. Now, for all we have,where,Now,Then,But implies as and . Hence which contradicts our assumption. Consequently, our claim is true and .
Therefore, we get , , .
Case 2. , , .
For and we can easily prove that and (as in Case 1).
For , we claim . If we assume that the claim is not true, then there exist such that and for all where and as , are the respective carrying capacities of patch 1 and patch 2. Now, for all we have,where,Now,Then,But implies as and . Hence which contradicts our assumption. Consequently, our claim is true and .
Therefore, we get , , .
Case 3. , , .
For and we have and (as proved in Case 1).
For let us claim that . Assume that the claim is false. Therefore, there exist and in such a way that as well as for all where and for all as , are the respective carrying capacities of patch 1 and patch 3.
Now, for all we have,where,Now,Therefore,But for all , as , and , .
Thus, which is a contradiction and proves that the claim is valid. Hence .
Therefore, in this case we also get , , .
Case 4. , , .
For we can prove that using the method of Case 3. Again for and we get and (from Case 1).
Therefore, we get .
Case 5. , , .
For we have (from Case 1) and we get for (from Case 3). Again gives (from Case 2).
Therefore, we get , , .
Case 6. , , .
For we can prove that (as in Case 3). In a similar way, considering Case 1, we get for . Lastly, for we can prove (from Case 2).
Consequently, we get , .
Case 7. , , .
For and , we can prove and (using Case 3). Again, for , we get (using Case 1).
Therefore, we get .
Case 8. , , .
Here we claim .
Let us assume that . Then for all .
Now taking .
we haveNow, , are the respective carrying capacities of patch 2 and patch 3. So, and (carrying capacity means the maximum population that can be achieved by the environment). Thus, as .
Again, the first equation of (1) gives,Also, we have, where .
This implies that which contradicts the assumption and so .
Similarly, we can prove that, .
Lastly let us assume that . Then for all .
Now taking , we have,Now, , are the respective carrying capacities of patch 1 and patch 2. So, and (carrying capacity means the maximum population that can be achieved by the environment). Thus as .
Again, the third equation of (1) gives,We also have, from the third equation of (1). So, . This again contradicts the assumption and we get .
Therefore, we conclude .
Let where for . Also let for ; and .
Then,Now, if we take , then the value of is maximum at and .
Therefore,This implies,Now applying the theory of differential inequality, we get,Taking ,Then all the solutions of (1) that initiate in are confined in the region: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 and for the respective cases.
Case 9. .
Clearly, the non-trivial equilibrium is a positive solution of the following system of equations:From equations (37), and (39) we have . Also, from (34) we obtain,Thus,