Abstract
This paper presents a three-dimensional continuous time dynamical system of three species, two of which are competing preys and one is a predator. We also assume that during predation, the members of both teams of preys help each other and the rate of predation of both teams is different. The interaction between prey and predator is assumed to be governed by a Holling type II functional response and discrete type gestation delay of the predator for consumption of the prey. In this work, we establish the local asymptotic stability of various equilibrium points to understand the dynamics of the model system. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system, and global stability of the positive interior equilibrium solution are discussed by constructing suitable Lyapunov functions when the gestation delay is zero, and there is no periodic orbit within the interior of the first quadrant of state space around the interior equilibrium. As we introduced time delay due to the gestation of the predator, we also discuss the stability of the delayed model. It is observed that the existence of stability switching occurs around the interior equilibrium point as the gestation delay increases through a certain critical threshold. Here, a phenomenon of Hopf bifurcation occurs, and a stable limit cycle corresponding to the periodic solution of the system is also observed. This study reveals that the delay is taken as a bifurcation parameter and also plays a significant role for the stability of the proposed model. Computer simulations of numerical examples are given to explain our proposed model. We have also addressed critically the biological implications of our analytical findings with proper numerical examples.
1. Introduction
In population dynamics, prey-predator coordination contributes a decisive responsibility through the last few decades [1, 2]. The dynamical relationship between predators and their preys has been recognized as an important topic in theoretical ecology since the discovery of the famous Lotka–Volterra equation. In ecosystem, prey-predator relationship contributes an imperative role. During the first World War, Lotka and Volterra symbolized mathematical appearance of prey-predator system [2, 3]. Since then, a huge number of research works have been carried out by following their mathematical expression. Hou [4] considered the permeance for general Lotka–Volterra model along with time delay, cooperation, and competition. Pal et al. [5] studied one-prey and one-predator harvesting system with the imprecise biological parameters. The nonautonomous Lotka–Volterra competition model is presented by Ahmed [6] and May [7] who discussed some simple mathematical models along with some complicated dynamics. There are a few approaches to signify prey-predator relations, for instance competition [8] and cooperation [3].
The stability and bifurcation investigation of prey-predator structure are determined by the functional response. In modelling the prey-predator structure, functional response makes a crucial contribution. There are several categories of functional responses in the subsistence literature [1, 2]. Holling category I functional response is categorized mathematically via a straight line through the origin [9, 10]. In the same way, the mathematical expression of Holling type II function is specified by where , , and have their respective meanings [9, 11–13].
A huge number of ecologists have premeditated prey-predator scheme by means of Holling category I functional response. In analysing the ecosystem, researchers have traced more information on two-dimensional prey-predator structure used for an elongated time. Food chain dynamics was discussed by Kuznetsov et al. [14] and Li et al. [15]. Srinivasu et al. [16] studied the consequences of assigning additional food on the dynamical system. Bandyopadhyay et al. [17] elucidated the dynamics of autotroph-herbivore ecosystem along with nutrient recycling. Theoretical ecologists were avoiding three or more dimensional species model system for an elongated time. It is mainly because higher dimensional models incorporate a greater number of differential equations which make it tricky to study the model structure. However, in a real ecosystem, higher dimensional models are very much imperative. Consequently, especially three-dimensional models are becoming more significant in different branches of ecology and ecosystem. Erbe et al. deals with the three-dimensional food chain model where mutual interference among predators and time delay due to gestation are proposed [18]. Fredman et al. presented a competition model involving three species [19]. The dynamical behavior of mussel and fish population is explained by Gazi et al. [20], and Maiti et al. [21] discussed the tritrophic food chain system with discrete time lag. Maiti et al. [22] extended the work and studied the effectiveness of biocontrol of pests in tea plants. Pal et al. [23] studied the influence of uncertainties in a food chain system. Pal et al. [24] presented a one predator and two prey systems by using fuzzy number and interval as biological parameters. The dynamical behavior of a one predator and two prey systems along with predator harvesting is studied by Gakkhar et al. [25].
Stage-structure-based prey-predator demonstrations by way of gestation time lag due to adulthood of the species are ornately discussed by numerous researchers. Bifurcation analysis of predator-prey models with the time lag is elucidated by Pal et al. [26] and Zhang [27]. Prey-predator system with discrete time lag and harvesting of the predator species is studied by Misra et al. [28], and stage-structured system of prey-predator with time lag for gestation is presented by Bandyopadhyay et al. [29]. Freedman et al. [30] presented Gauss prey-predator system including mutual interference and gestation time lag. This type of depiction is ended by inserting time delay in the differential equation. Generally, when predator species munch through the prey species, alteration of prey biomass into predator biomass is not instant. This necessitated some time lag for the alteration. Consequently, prey-predator-based holdup coordination is very much indispensable in mathematical ecology.
Naik et al. [31] recently introduced a two-dimensional discrete time chemical model with the subsistence of its fixed points; along with this, the flip and generalised flip bifurcations are identified for this system. The 1- and 2-parameter bifurcations of discrete time predator-prey model with the mixed functional response are discussed by Naik et al. [32]. Naik et al. [33] investigated the complex dynamical aspects of discrete-time Bazykin–Berezovskaya predator-prey system along with strong Allee effect.
A three-dimensional prey-predator model where two prey groups help each other from the predator group is discussed by Elettreby [3] and Tripathi et al. [34, 35] extended the work adding the competitive interaction among prey groups when there is no predator group present.
Following the works of Elettreby [3] and Tripathi [34, 35], in this contemporary circumstance, we deem a three-dimensional prey-predator (two prey teams and one predator team) structure with help and discrete type gestation delay of the predator. Holling type II functional rejoinder is used for interface amid prey squads and predator squad. In nonappearance of predator, the prey teams fight with apiece other for widespread food wherewithal. Once more, when prey teams are assaulted by the predator, then two prey species help each other for defensing them from predator. Also, after chomp through the prey, the escalation of predator species is not immediate, and it requires some time insulate for the exchange.
In this paper, we have discussed a three-dimensional predator-prey model with logistic equation where the prey species are competing with each other for the essential elements, e.g., food and space, and also, two teams of prey species are helping each other at the time of predation. To the best of our knowledge, all these above factors, at the same time, have not yet used.
Rest of the paper is presented in the following manner: research gaps are presented in Section 2. Mathematical portrayal of our projected structure is carried out in Section 3. Section 4 presents the positivity, boundedness, and permanence of our planned model. Behavior of the model in nonappearance of delay is described in Section 5. Behavior of our planned model in presence of delay is depicted through Section 6. Numerical illustrations through graphical staging are presented in Section 7. General discussion about our proposed model system is conducted in Section 8. Concluding remark is delivered in Section 9.
2. Research Gaps
Different types of prey-predator models along with different types of factors are analysed by many researchers. To discuss their work in a simplified way, we have presented a table which briefly explains the work carried out till now. In this table, the comparative discussion has been carried out in a tabular form which gives a quick overview about the research gaps. We have categorised the work on six main terms, viz., competition, mutualism, time delay, one predator-two preys, logistic equation, and Holling type-II functional response. From Table 1, it is clear that all the factors have never been used at the same time by any researchers, but all these conditions are used in our proposed model.
3. Mathematical Portrayal of the Model Structure
Our anticipated mathematical model is supported in the subsequent suppositions:(i)In nonexistence of the predator, both the preys are budding logistically(ii)In nonattendance of the predator, two teams of preys fight with each other for widespread wherewithal(iii)Two teams of preys are plateful themselves for the fortification from their attackers(iv)Prey populace augmentation rate is abridged due to the consequence of predation which is deliberate by a term comparative to the prey and predator populations(v)Predator’s death is cropped up due to nonappearance of any prey teams(vi)There might be antagonism amid the predator individuals due to insufficient quantity of food supply(vii)For the development of predators, a time lag is assumed(viii)Our wished-for model is reserved only by two preys and one widespread predator
According to the suppositions (i)–(viii), our anticipated model structure can be articulated mathematically in the subsequent approach.by means of the preliminary conditionswhere denotes the population density of the first prey, denotes that of the second prey, and denotes the population density of predator species; and are environmental carrying capacity of prey species and , respectively; and put up intrinsic augmentation rates of and correspondingly; and correspond to the per capita decrease rate of and correspondingly; and give the environment defense for the species and , respectively; and stand for the effect of handing time for predators; , and , denote the competition rates in the absence of predator species and cooperation coefficients for the prey species and , respectively; , , , and stand for natural death rate of predator; , density dependence rate of the predator, exchange rate of and into new offspring of predator species, respectively; designates the requisite time taken by the prey species to become an adult. Finally, we consider that the coefficients , , , , , , , , , , , , , , , , , and are all positive numbers.
Our wished-for mathematical model is fine fitted for the group of Gazelles and Zebra which serve as two teams of prey and their attacker Tiger or Lion play the task of predator.
4. Positivity, Boundedness, and Permanence of Our Anticipated Model
The intention of this segment is to confer about the positivity, boundedness, and permanence of our anticipated representation (1) amid preliminary settings (2). To ascertain the affirmed behavior of our wished-for model, we state a foremost Lemma.
Lemma 1. Under the conditions , , and , , afterward .
Consequent Hale [46] and Jordan [47], we affirm the subsequent theorem.
Theorem 2. The coefficients , , , , , , , , , , , , , , , , and are bounded positive quantities. Subsequently, the model structure (1) has a sole solution on by means of opening conditions (2).
Theorem 3. Solutions of the model scheme (1) through preliminary conditions (2) are always greater than zero for all positive values of .
Proof. We stumble on the fact that the right hand side of the model system (1) is absolutely continuous in addition to locally Lipschitzian on the space of continuous functions. Therefore, the solution of (1) by way of primary conditions (2) subsists and is unique on for all . Equation of one of the model systems (1) givesNext equation of the model scheme (1) providesIn the same way, the last equation of the model structure (1) affordswhich concludes the proof of the theorem.
Theorem 4. Let where , , , and . As a result of that, is invariantly positive.
Proof. If it is assumed that , then is always positive. If we can prove that , , and , then it is clear that for all values of greater than or equal to zero. In the first attempt, we try to prove that . For avoiding the population outburst, it is assumed that the assistance term is dominated by competition amid prey species as well as interface among prey and predator species . Using the said deliberation and positive values of , , and , the first equation of the model scheme (1) bestows thatFrom (6), we get for all values of greater than or equal to zero. Again, next equation of (1) bestows thatAgain from (7), we have for all values greater than or equal to zero. Also, the third equation of (1) bestows thatUsing (8), we have for all values of greater than or equal to zero, where . This completes the proof.
Theorem 5. If the conditions , , and are contented, then the model scheme (1) is permanent. , , , and are defined in the proof given underneath.
Proof. Due to amply bulky , equation (6) provides . Furthermore, for amply bulky , equations (7) and (8) provide and . As , , and are positive, the first equation of (6) furnishesdue to amply bulky , where . If , i.e., is satisfied, then Lemma 1 providesThus, for any arbitrary value of , there exists a number such that for all values of greater than . In a similar fashion, second equation of (1) providesfor sufficiently large , where . If , i.e., is satisfied, then Lemma 1 givesHence, for any arbitrary , there exists a number such that for all values of greater than . Again, last equation of (1) providesAs an upshot of arbitrary and , the above differential inequality can be expressed aswhere , , and . If the condition , i.e., is fulfilled, then Lemma 1 providesfor sufficiently large . Also, from inequalities (6)–(8), together with Lemma 1, we haveNow, choosing and , we obtain the permanence of the system (1).
5. Model Structure with Nonappearance of Time Lag
Model system (1) captures the subsequent structure in nonappearance of time lag together with preliminary stipulations
5.1. Subsistence of Equilibrium Points and Local Stability Investigation
The probable equilibrium points are specified underneath:
.
It is palpable that the equilibrium points , , and subsist forever. Our only task is to authenticate the existence of lingering equilibrium points.
5.1.1. Subsistence of
By solving the first two linear simultaneous equationswe get and . Therefore, exists provided and .
Remark 6. If is considered, then and are equal. Again if is satisfied, then and are the same.
5.1.2. Subsistence of
Consider two nonlinear equations
From (21), we get
Putting the value of in (20), we havewhere , , , and . For the positive and unique solution of the (23), the stipulations specified underneath must be satisfied.
Therefore, the equilibrium point subsists if the above supposed stipulations (24) are fulfilled.
Remark 7. In the same way, we easily prove that equilibrium point subsists under the conditions and .
5.1.3. Subsistence of
Clearly, is achieved by solving the set of nonlinear simultaneous equations prearranged by
Solving (25) and (27), we havewhere . From (28), if , then , wherewhere , , , and . As ; therefore, the (29) has a positive solution if , , and .
From (28), one can obtain
It is evident that if either
From (26), we calculate and substitute it in (27), and we acquire
From (32), one can observe that, when , then , wherewhere , , and . Since , the (33) has a positive solution if and .
From (32), one can obtain that
It is obvious that if either
Therefore, the meeting point of (28) and (29) is unique. Also, the conditions (31) and (35) and the inequality are fulfilled. Again, by placing the values of and in (27), we have achieved the value of . So, the subsistence of positive inner equilibrium point is verified.
At present, we are in the situation to talk about the local stability behavior of the model structure (17) at each proposed equilibrium points.
Theorem 8. Nature of equilibrium point is saddle point.
Proof. At the equilibrium point , the variational matrix of the model structure (17) has obtained the formThe eigenvalues of are , , and and , , and . Therefore, is the saddle point in nature in conjunction with unstable manifold in and directions, respectively, and stable manifold in the direction.
Theorem 9. If the conditions and are satisfied, then axial equilibrium point is stable in nature.
Proof. At the equilibrium point , the variational matrix of the model structure (17) takes the formThe eigenvalues of are , , and . Now if and , i.e., if and , then is stable in nature.
Theorem 10. is stable if and
Proof. In the same way as above, this theorem can be proved.
Remark 11. From the previous three theorems, ecologically it can be interpreted that the co-operating coefficients and do not give any participation for establishing the stability behavior of , , and . The intercompetition coefficients and and the interference coefficients and provide positive effect on the stability behavior of and correspondingly.
Theorem 12. The predator-free equilibrium point is stable if the conditions and are contented.
Proof. Analogous to , one of eigenvalues of the matrix is specified by . Since is a 3 × 3 matrix, the remaining two eigenvalues are the solutions of the following equation:where and . As , according to Routh–Hurwitz criterion [1], the (38) has negative real part solutions if , i.e., . Hence, if the conditions and are contented, then is stable in nature.
At , matrix takes the formwhere , , , , , , , , and .
Therefore, the characteristic equation of iswhere , , and .
Therefore, using Routh–Hurwitz criterion [1, 2], we say that the solutions of the characteristic (40) has real component with less than zero iff
Theorem 13. If the provision (41) is fulfilled, then is stable in nature.
Over again, at the point , obtains the formwhere , , , , , , , , and .
Therefore, the characteristic equation of iswhere , , and .
Using Routh–Hurwitz condition [1, 2], we obtain the solutions of the characteristic (43) has nonpositive real component iff
Theorem 14. If the stipulation (44) is fulfilled, then is stable in nature.
Finally, at , takes the formwhere , , , , , , , , and .
Therefore, the characteristic equation of iswhere , and .
Again using Routh–Hurwitz condition [1, 2], we obtain the solutions of the characteristic (46) has nonpositive real component iff
Theorem 15. If the conditions (47) are contented, then inner equilibrium point is stable in nature.
5.2. Stability in Global Perspective
The current section provides stability performance of the model structure (17) at in a global point of view.
Theorem 16. The conditions (A.1) and (A.2) imply the global stability behavior of the equilibrium point .
Proof. See Appendix.
6. Model Analysis due to Time Lag
Due to time lag , stability nature of our wished-for replica structure (1) at is offered in the contemporary part. At , the system (1) has the characteristic equation as given in the following equation:where , , , , , , , , , , , , , and .
Let , be a solution of (48). Therefore, it is evident that
Separating real and imaginary parts of (49), we attain
Adding both squared equations of (50), the subsequent equation is obtainedwhere , , and . Therefore, the sole positive root of (51) is obtained under the conditions , , and . So, we dig up a pair of imaginary solutions of (48). The value of is gettable by substituting the value of in (50). The idiom of is specified by
Next, Lemma is followed by the over argument.
Lemma 17. The couple of imaginary solutions of (48) is attained for .
Theorem 18. Assume is defined by (52), also subsists. , , and as well as the conditions (47) are satisfied. If increases through zero, then there exists a value of say for which is asymptotically stable for , and it becomes unstable when . In addition to that, for (where for ) at the point , the structure (1) experiences a Hopf bifurcation.
Proof. For , by the stipulation (47), the inner equilibrium point is stable in nature. Therefore, using Butler’s lemma [30], we obtain that the inner equilibrium point remains stable under the condition . Our main intention is to show that the value of is always greater than zero, which implies that when , our proposed structure has the slightest one positive eigenvalue with positive real component. Depending on the above discussion, we conclude that the conditions of Hopf bifurcation are fulfilled along with expected periodic solution. Differentiating both sides of (48) with regard to , we achieveFrom (53), we getThus, . After some algebraic manipulations from (55), we getHence, if is greater than zero and is less than zero. Hence, the transversality stipulation is fulfilled and Hopf bifurcation arose at and . Thus, the proof of the theorem is completed.
6.1. Assessment of the Time Lag Length to Safeguard Stability
To estimate the time lag length to protect the stability of the model structure (1), we initialized the system concerning its inner equilibrium point . The initial model structure is prearranged underneathwhere , , and .
If we apply Laplace transform on both sides of (57), we eventually gain
Here, and . The Laplace transform of , , and is expressed by , , and correspondingly.
Using Nyquist theorem [26] and from [18], the local asymptotic stability stipulations of the inner equilibrium point can be articulated in the subsequent form.
Here, . The smallest positive solution of the (60) is .
In the previous section, we detect that in nonappearance of time lag, inner equilibrium point is stable. Then, by Bulter’s lemma [18], we have adequately tiny , and all eigenvalues will be negative real components. Also, when augments through zero, one can assure that there are no eigenvalues with positive real component that bifurcates from infinity.
In this current circumstance, stipulations (59) and (52) bestow
If the conditions (61) and (62) are fulfilled concurrently, then these stipulations furnish the sufficient stipulations for assurance stability. We shall utilize them to get an estimate on the length of delay. To estimate the length of time lag, we shall use these stipulations. Our objective is to specify the upper bound of which is independent of in such a way that (61) holds for all values of where . Therefore, for a particular value of say , we can redraft (62) as follows:
Maximizing subject to and , we obtain
Hence, ifit is clear from (65) that .
Again (55) provides
Also, for , the inner equilibrium point is stable as well as (66) holds due to adequately tiny . Replacing (63) into (66) gives
The bounds of providesand
Now, from (67)–(69), we getwhere , , and .
Hence, ifthen stability is preserved for .
From the above discussed outcomes, the next theorem is followed.
Theorem 19. If the time lag satisfies the inequality , then the model structure (1) is locally asymptotically stable where is provided in (71).
7. Numerical Verifications
Numerical verification of analytical findings is very much important from a practical view point. This verification is not possible without the help of a computer software like MATLAB and Mathematica. In this current section, we have mainly verified the analytical finding by graphical presentation. Authentication of analytical finding of the model structure (1) is very much significant from a realistic standpoint also.
For the model structure (17), the values of the biological parameters and initial populations are taken in the subsequent way , , , , , , , , , , , , , , , , , , and .
For our setting parameter values, the inner equilibrium point is equal to . Also, for this set of parameter values, all the species persist and we get a nontrivial equilibrium point . At , conditions of Theorem 15 are fulfilled as , , and . Hence, is locally asymptotically stable.
Figure 1 depicts that starting with an initial condition , the populations reach their respective stable situation , , and in a limited period of time.
As per our expectation, we monitor from Figure 1 that as the predator species steadily enlarges and both the prey species steadily diminishes and finally after a limited period of time, the population system comes to a steady-state situation.
Figures 2, 3, and 4 present the plane, plane, and plane protrusions of the system (17) correspondingly.
Figure 2 illustrates that in the projection, the trajectory starting with the initial condition converges to the inner equilibrium point . Similarly, Figures 3 and 4 portray the plane and plane projection, respectively. In Figures 3 and 4, the trajectory starting with the initial condition converges to the inner equilibrium point .
Next, we investigate for the delay model (1). It is a well-known fact that if a model structure is stable in nonattendance of time lag (), it is not assured that the system remains stable in the occurrence of time lag . Let us choose the parametric values of the same system as stated above. Now, for these choices of parameters, Theorem 18 and Lemma 17 assured that (51) has a sole positive solution and (52) gives the critical value . Using Theorem 18 and Figures 5(a), 5(b) and 6(a), 6(b), it is monitored that when , then the inner equilibrium point exhibits asymptotically stable behavior.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
Figure 5 depicts that as , all the species of the population system converges to their respective stable state levels also as and if all the other parameter values are kept the same as stated above, then a delayed model structure becomes unstable. Again from Figure 6, the phase space diagram is portrayed.Therefore, when we augment the values of above the critical value , the population system exhibits growing oscillatory behavior. From Figures 5 and 6, the change in the stability behavior of this system is clearly visible. When the value of is slightly higher than its critical value , the stable equilibrium point becomes unstable.
Therefore, we may conclude that keeping other parameters fixed, if we take , then becomes unstable and exhibits Hopf bifurcation, and a bifurcating periodic solution is noticed around .
8. Results and Discussions
The current paper studied a three-dimensional prey-predator co-operative structure along with gestational time lag of the predator species. In the ecosystem, there exist many species who lived in a crowd and cooperate themselves by distributing similar territory. As the species sharing the same territory, depending on proper circumstances, the grouped populations may co-operate sometimes, and they may also sometimes compete with themselves. Sea anemone and the clown fish are the good examples of the grouping population. A proper predator-prey structure can be formed with the help of these categories of grouping species. For the existence of a particular species, the most essential elements are food and space. Therefore, the prey species are competing with each other for such kind of general assets. On the other hand, predator species cropped the prey population at a fixed rate due to their survival. Also, the alteration of prey biomass to predator biomass is not instantaneous; it needs some time lag for alteration. Motivated by these facts, in this current paper, we create and investigate dissimilar behaviors of a prey-predator time lag model structure consisting of two groups of contending as well as supportive preys and one group of predators.
The existences of different equilibrium points of the model structure (1) and their stabilities are pointed out carefully. Global stability behavior of the inner equilibrium point is addressed properly. We finally observe that when the delay parameter (critical vale of ), the stability nature of the inner equilibrium point becomes unstable and exhibits Hopf bifurcations. Our analytical findings are properly illustrated graphically through Figures 1 to 6 correspondingly. Time-series plot of , , , plane projection, plane projection, and plane projection is described, and the stable and unstable phase space diagrams are illustrated in the above figures. Also, it is demonstrated that the phase portrait of the model is stable for and unstable for .
9. Conclusions
Finally, we conclude that the whole of our proposed delayed model structure is supposed in a deterministic environment. However, our model can be made more pragmatic and attractive if it is supposed in fuzzy, interval, or in stochastic environment for some parameter uncertainties or some other environmental characteristics. This conception is left for further research trend. As a part of future work, to make the system more realistic, we can include impreciseness in the parameters of the model to enhance our model.
Appendix
Let us define a Lyapunov function asfor the positive values of the constants , , and which will be specified soon after.
At this point, since for and . Differentiating with regard to alongside, the solutions of model (17) provide
By putting , , and , subsequently making simpler , we acquired that
Clearly, at .
Now, ifor
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.