Analysis of Prey-Predator Scheme in Conjunction with Help and Gestation Delay

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Introduction
In population dynamics, prey-predator coordination contributes a decisive responsibility through the last few decades [1,2].Te 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 frst 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.Te nonautonomous Lotka-Volterra competition model is presented by Ahmed [6] and May [7] who discussed some simple mathematical models along with some complicated dynamics.Tere are a few approaches to signify prey-predator relations, for instance competition [8] and cooperation [3].
Te stability and bifurcation investigation of preypredator structure are determined by the functional response.In modelling the prey-predator structure, functional response makes a crucial contribution.Tere 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 specifed by ex(f + x) − 1 where e, x, and f have their respective meanings [9,[11][12][13].
A huge number of ecologists have premeditated preypredator 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.Teoretical 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 diferential 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 signifcant in diferent 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].Te dynamical behavior of mussel and fsh 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 efectiveness of biocontrol of pests in tea plants.Pal et al. [23] studied the infuence 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.Te 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.Tis type of depiction is ended by inserting time delay in the diferential equation.Generally, when predator species munch through the prey species, alteration of prey biomass into predator biomass is not instant.Tis 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 fxed points; along with this, the fip and generalised fip bifurcations are identifed for this system.Te 1-and 2parameter 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 efect.
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 fght 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.

Research Gaps
Diferent types of prey-predator models along with diferent types of factors are analysed by many researchers.To discuss their work in a simplifed way, we have presented a table which briefy 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 2 Journal of Mathematics factors have never been used at the same time by any researchers, but all these conditions are used in our proposed model.

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 fght with each other for widespread wherewithal (iii) Two teams of preys are plateful themselves for the fortifcation 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) Tere might be antagonism amid the predator individuals due to insufcient quantity of food supply (vii) For the development of predators, a time lag l 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 conditions where X p (t) denotes the population density of the frst prey, X q (t) denotes that of the second prey, and X r (t) denotes the population density of predator species; ξ 1 and ξ 2 are environmental carrying capacity of prey species X p and X q , respectively; ζ 1 and 8 1 put up intrinsic augmentation rates of X p and X q correspondingly; ζ 2 and 8 2 correspond to the per capita decrease rate of X p and X q correspondingly; μ and ρ give the environment defense for the species X p and X q , respectively; μ 1 and ρ 1 stand for the efect of handing time for predators; ζ 3 , 8 3 and ζ 4 , 8 4 denote the competition rates in the absence of predator species and cooperation coefcients for the prey species X p and X q , respectively; ϕ 1 , ϕ 2 , ϕ 3 , and ϕ 4 stand for natural death rate of predator; X r , density dependence rate of the predator, exchange rate of X p and X q into new ofspring of predator species, respectively; l ≥ 0 designates the requisite time taken by the prey species to become an adult.
Our wished-for mathematical model is fne ftted 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.

Positivity, Boundedness, and Permanence of Our Anticipated Model
Te 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 afrmed behavior of our wished-for model, we state a foremost Lemma.

Theorem . Te coefcients ζ
μ, ρ, μ 1 , and ρ 1 are bounded positive quantities.Subsequently, the model structure (1) has a sole solution on [0, +∞) 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 t.
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.Terefore, the solution (X p (t), X q (t), X r (t)) of ( 1) by way of primary conditions (2) subsists and is unique on [0, ζ) for all ζ ∈ (0, +∞).Equation of one of the model systems (1) gives Next equation of the model scheme (1) provides In the same way, the last equation of the model structure (1) afords which concludes the proof of the theorem.
and X r (t) ≤ Z 3 , then it is clear that (X p (t), X q (t), X r (t)) ∈ Q for all values of t greater than or equal to zero.In the frst attempt, we try to prove that X p (t) ≤ Z 1 .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 X p and predator species X r .Using the said deliberation and positive values of X p , X q , and X r , the frst equation of the model scheme (1) bestows that From ( 6), we get X p (t) � max X p (0), ξ 1   � Z 1 for all values of greater than or equal to zero.Again, next equation of (1) bestows that Again from (7), we have X q (t) � max X q (0), ξ 2   � Z 2 for all values greater than or equal to zero.Also, the third equation of (1) bestows that Using ( 8), we have X r (t) ≤ max X r (0), ξ 3   � Z 3 for all values of greater than or equal to zero, where are contented, then the model scheme ( 1) is permanent.M, N, f 1 , and f 2 are defned in the proof given underneath.

Model Structure with Nonappearance of Time Lag
Model system (1) captures the subsequent structure in nonappearance of time lag l together with preliminary stipulations X p (0) > 0, X q (0) > 0 as well as X r (0) > 0. (18)

Subsistence of Equilibrium Points and Local Stability
Investigation.Te probable equilibrium points are specifed underneath: It is palpable that the equilibrium points Γ 1 , Γ 2 , and Γ 3 subsist forever.Our only task is to authenticate the existence of lingering equilibrium points.

Subsistence of Γ 4 . By solving the frst two linear simultaneous equations
From ( 21), we get Putting the value of X q ′ in (20), we have where For the positive and unique solution of the (23), the stipulations specifed underneath must be satisfed.

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Journal of Mathematics Terefore, the equilibrium point subsists Γ 5 if the above supposed stipulations (24) are fulflled.
From (28), one can obtain From ( 26), we calculate X * r and substitute it in (27), and we acquire From (32), one can observe that, when X * p ⟶ 0, then X * q ⟶ X * qb , where where From (32), one can obtain that It is obvious that dX * q /dX * p < 0 if either n 2 > 0 and m 2 > 0 or n 2 < 0 and m 2 < 0. ( Terefore, the meeting point of ( 28) and ( 29) is unique.Also, the conditions (31) and (35) and the inequality X * qa < X * qb are fulflled.Again, by placing the values of X * p and X * q in (27), we have achieved the value of X * r .So, the subsistence of positive inner equilibrium point Γ 7 is verifed.
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 Γ 1 (0, 0, 0) is saddle point.
Proof.At the equilibrium point Γ 2 (ξ 1 , 0, 0), the variational matrix J Γ 2 of the model structure (17) takes the form Journal of Mathematics Te eigenvalues of 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 coefcients ζ 4 and 8 4 do not give any participation for establishing the stability behavior of Γ 1 , Γ 2 , and Γ 3 .Te intercompetition coefcients 8 3 and ζ 3 and the interference coefcients μ 1 and ρ 1 provide positive efect on the stability behavior of Γ 2 and Γ 3 correspondingly.
Over again, at the point Γ 6 (X p ″ , 0, X r ″ ), J Γ 6 obtains the form where Terefore, the characteristic equation of J Γ 6 is where Using Routh-Hurwitz condition [1,2], we obtain the solutions of the characteristic (43) has nonpositive real component if Theorem 14.If the stipulation ( 44) is fulflled, then Γ 6 (X p ″ , 0, X r ″ ) is stable in nature.

Model Analysis due to Time Lag
Due to time lag (≠0), stability nature of our wished-for replica structure (1) at Γ 7 (X * p , X * q , X * r ) is ofered in the contemporary part.At Γ 7 (X * p , X * q , X * r ), the system (1) has the characteristic equation as given in the following equation: where Let λ � ik, (k > 0) be a solution of (48).Terefore, it is evident that Separating real and imaginary parts of (49), we attain Adding both squared equations of (50), the subsequent equation is obtained where

and
Terefore, the sole positive root k 2 + of ( 51) is obtained under the conditions Q 1 > 0, Q 2 > 0, and Q 3 < 0. So, we dig up a pair of imaginary solutions ±ik + of (48).Te value of l is gettable by substituting the value of k 2 + in (50).Te idiom of l is specifed by Next, Lemma is followed by the over argument.
Lemma 17. Te couple of imaginary solutions of ( 48) is attained for l � l + 0 .
Proof.For l � 0, by the stipulation (47), the inner equilibrium point Γ 7 (X * p , X * q , X * r ) is stable in nature.Terefore, using Butler's lemma [30], we obtain that the inner equilibrium point Γ 7 (X * p , X * q , X * r ) remains stable under the condition l < l + 0 .Our main intention is to show that the value of d(Reλ)/dl| l�l + 0 ,k�k + is always greater than zero, which implies that when l > l + 0 , 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 fulflled along with expected periodic solution.Diferentiating both sides of (48) with regard to l, we achieve

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From (53), we get .After some algebraic manipulations from (55), we get greater than zero and Q 3 is less than zero.Hence, the transversality stipulation is fulflled and Hopf bifurcation arose at k � k + and l � l + 0 .Tus, the proof of the theorem is completed.□

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 Γ 7 (X * p , X * q , X * r ).Te initial model structure is prearranged underneath where z 1 (t) � X p (t) + X * p , z 2 (t) � X q (t) + X * q , and z 3 (t) � X r (t) + X * r .If we apply Laplace transform on both sides of (57), we eventually gain Journal of Mathematics Here, σ 1 (r) �  0 − l e − rt z 1 (t)dt and σ 2 (r) �  0 − l e − rt z 2 (t)dt.Te Laplace transform of z 1 (t), z 2 (t), and z 3 (t) is expressed by z 1 (r), z 2 (r), and z 3 (r) correspondingly.
Using Nyquist theorem [26] and from [18], the local asymptotic stability stipulations of the inner equilibrium point Γ 7 (X * p , X * q , X * r ) can be articulated in the subsequent form.
In the previous section, we detect that in nonappearance of time lag, inner equilibrium point Γ 7 (X * p , X * q , X * r ) is stable.Ten, by Bulter's lemma [18], we have adequately tiny >0, and all eigenvalues will be negative real components.Also, when l augments through zero, one can assure that there are no eigenvalues with positive real component that bifurcates from infnity.
In this current circumstance, stipulations (59) and ( 52) bestow If the conditions (61) and (62) are fulflled concurrently, then these stipulations furnish the sufcient 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 w + of w 0 which is independent of l in such a way that (61) holds for all values of w where w ∈ [0, w + ].Terefore, for a particular value of w say w 0 , we can redraft (62) as follows: Maximizing Ω 3 + Ω 5 cos w 0 l + Ω 4 w 0 sin w 0 l subject to |sin w 0 l| ≤ 1 and |cos w 0 l| ≤ 1, we obtain Hence, if it is clear from (65) that w 0 ≤ w + .Again (55) provides Also, for l � 0, the inner equilibrium point Γ 7 (X * p , X * q , X * r ) is stable as well as (66) holds due to adequately tiny l > 0. Replacing (63) into (66) gives Te bounds of w 0 provides and Now, from (67)-(69), we get where Hence, if then stability is preserved for 0 ≤ l < l + .
From the above discussed outcomes, the next theorem is followed.

Theorem 19.
If the time lag l satisfes the inequality 0 < l < l + , then the model structure ( 1) is locally asymptotically stable where l + is provided in (71).

Numerical Verifications
Numerical verifcation of analytical fndings is very much important from a practical view point.Tis verifcation is not possible without the help of a computer software like MATLAB and Mathematica.In this current section, we have mainly verifed the analytical fnding by graphical presentation.Authentication of analytical fnding of the model Journal of Mathematics structure (1) is very much signifcant from a realistic standpoint also.
Figure 1 depicts that starting with an initial condition (0.001, 0.05, 0.008), the populations reach their respective stable situation X * p , X * q , and X * r 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 fnally after a limited period of time, the population system comes to a steady-state situation.
Figures 2, 3, and 4 present the X p X q plane, X p X r plane, and X q X r plane protrusions of the system (17) correspondingly.
Figure 2 illustrates that in the X p X q projection, the trajectory starting with the initial condition (0.001, 0.05, 0.008) converges to the inner equilibrium point Γ 7 .Similarly, Figures 3 and 4 portray the X p X r plane and X q X r plane projection, respectively.In Figures 3 and 4, the trajectory starting with the initial condition (0.001, 0.05, 0.008) converges to the inner equilibrium point Γ 7 .
Next, we investigate for the delay model (1).It is a wellknown fact that if a model structure is stable in nonattendance of time lag (l � 0), it is not assured that the system remains stable in the occurrence of time lag (l ≠ 0).Let us choose the parametric values of the same system as stated above.Now, for these choices of parameters, Teorem 18 and Lemma 17 assured that (51) has a sole positive solution k + � 0.0417958 and (52) gives the critical value l + 0 � 23.7602.Using Teorem 18 and Figures 5(a Figure 4: X q X r plane projection of the solution with X p (0) � 0.001, X q (0) � 0.05, X r (0) � 0.008.

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Figure 5 depicts that as l < 23.7602, all the species of the population system converges to their respective stable state levels also as l > 23.7602 � l + 0 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.Terefore, when we augment the values of l above the critical value l + 0 , 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 l is slightly higher than its critical value l + 0 , the stable equilibrium point becomes unstable.

Results and Discussions
Te current paper studied a three-dimensional preypredator 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 fsh are the good examples of the grouping population.A proper predator-prey structure can be formed  1) with X p (0) � 0.51, X q (0) � 0.42, X r (0) � 0.3.(a) Stable behavior of X p , X q , X r for l � 21 < l + 0 � 23.7602,(b) stable behavior of X p , X q , X r for l � 23 < l + 0 � 23.7602, (c) unstable behavior of X p , X q , X r for l � 24 > l + 0 � 23.7602, and (d) unstable behavior of X p , X q , X r for l � 28 > l + 0 � 23.7602.
14 Journal of Mathematics with the help of these categories of grouping species.For the existence of a particular species, the most essential elements are food and space.Terefore, 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 fxed 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.Te existences of diferent 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 fnally observe that when the delay parameter l < l + 0 (critical vale of l ), the stability nature of the inner equilibrium point becomes unstable and exhibits Hopf bifurcations.Our analytical fndings are properly illustrated graphically through Figures 1 to 6 correspondingly.Time-series plot of X p , X q , X r , X p X q plane projection, X p X r plane projection, and X q X r plane projection is described, and the stable and unstable phase space diagrams are illustrated in the above fgures.Also, it is demonstrated that the phase portrait of the model is stable for l < 23.7602 and unstable for l > 23.7602.

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.Tis 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.1) with X p (0) � 0.51, X q (0) � 0.42, X r (0) � 0.93.(a) Stable behavior of X p , X q , X r for l � 21 < l + 0 � 23.7602,(b) stable behavior of X p , X q , X r for l � 23 < l + 0 � 23.7602, (c) unstable behavior of X p , X q , X r for l � 24 > l + 0 � 23.7602, and (d) unstable behavior of X p , X q , X r for l � 28 > l + 0 � 23.7602.

Table 1 :
Summary of the research gap based on literature review.