Analysis of Prey, Predator and Top Predator Model Involving Various Functional Responses

This research proposes a mathematical model to investigate the dynamical behavior of the system of three species, namely prey, predator and top predator. The feeding behavior of each predator serves as a functional response. The interaction between the species is carried out by a functional response. Crowley Martin functional response is incorporated between prey and predator while Holling type III functional response occurs between predator and top predator. The existence of positivity and boundedness of the system have been examined. The equilibrium points of the system are determined. The system has been linearized by applying the Jacobian matrix. The main perspective used to discuss the system's dynamics is that of permanence and stability. Further stability analysis of the system is carried out at around each equilibrium point. To comprehend the dynamics of the model system, the asymptotic stability of several equilibrium solutions, both local and global, is investigated . Routh Hurwitz criteria are used to analyze local stability at every equilibrium point. Using an appropriate Lyapunov function, the global asymptotic stability of the positive interior equilibrium solution is established. From a biological perspective, a system is considered to be permanent if all of its populations continue to exist in the future. The existence of permanence conditions of the system have been determined. To support the analytical results, several numerical simulations are carried out using the MATLAB software. Finally based on the results of the analytical and numerical simulations, the impact of the functional response between the prey, predator and top predator was discussed.


Introduction
In recent years, ecological modelling research has become more interesting to both mathematicians and biologists because of its dynamism.The richness of the dynamics is yielded by the interaction of the species in the ecology.Moreover, the interaction of the species is fascinating to investigate in the ecosystem.In 1798, Malthus formulated a singlespecies model.The modification of the single-species model was developed by Verhulst in 1838.Based on the single-species model many models were formulated.A two-species model was developed by Lotka and Volterra such as Prey and Predator in 1926 [1][2][3]  Baghdad Science Journal II, III, and IV to study the interaction between prey and predator employing harvesting, refuge, and these responses.The relationship between a predator's rate of prey consumption per unit of time and the quantity or density of its prey is known as the functional response 4 .Many researchers studied the dynamics of prey predator model in the presence of various functional response, Allee effect, reaction and diffusion 5,6 .Later on, three species and many species models were developed.In 1961, Kerner 7 expanded the Lotka-Volterra Model to include a three-species feeding chain.Chauvet et al 8 investigated a linear food chain for a three-species lotka-Volterra model.
Hasting and Powell 9 developed the three-level food chain model which is linear and demonstrates the chaotic dynamics in the ecosystem.The threespecies model's oscillatory behavior has been investigated 10 .Klebanoff created an ecological model class that exhibits the chaotic behavior of a three-species model with bifurcation 11 .A tri-trophic food chain model with a hybrid functional response was investigated for its chaotic behavior 12,13 .The dynamics of the fractional order prey-predator model were studied by Prabir Panja together with harvesting 14 .Zabidin Salleh et al 15

incorporated
Holling type III functional responses in the tritrophic food chain model.A cyclic three-species model's dynamic behavior was mathematically explained by Krishna das et al 16 .The dynamic behavior of the three-species food chain model was examined by numerous authors with various functional responses [17][18][19] .Ashok Mondal used the Crowley Martin functional response, which displays the Hopf bifurcation and persistence, to analyse the dynamical behavior of the food chain model 20 .
Permanence and persistence refer to each species' ability to persist over the long term in a given population and it was first introduced by Goodman

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. Many authors have looked into the longevity of the three-species model 22,23 .Arif et al 24 investigated the non-autonomous prey-predator model's reaction to the fluctuation rescue effect.Ali et al 25 studied the dynamics of food chain model involving Holling type IV and Holling type II functional response with leslie gower model.Naji 26 studied the chaotic dynamics of the prey-predator relationship.The chaotic behavior of food chain model with Holling type IV functional response was studied by Ali et al 27, 28 .The stability analysis of three species model with prey T axis has been investigated 29,30 .The behavior of three species model with the effect of noise has been examined 31,32 .Researchers have looked at the dynamic behavior of a three-species model including intraspecific rivalry between predators 33,34 .The three-species model's durability and stability were extensively researched by several authors [35][36][37] .Many studies have incorporated the dynamic interactions of a three-species model with diverse functional responses, Allee effects, interspecific competition, refuges, and various types of delays.The models previously used were based on either a single functional response or the same type of functional response.Now, they include a mixed functional response in the model.The model introduces a novel approach by combining a mixed functional response, including Holling type III and Crowley Martin functional responses, in a threespecies model.Crowley-Martin's functional response is a suitable choice among many functional responses.The Crowley-Martin functional response is utilized in situations where there is no predation occurring in a large population of both prey and predators.The ecosystem dynamics are more accurately represented by the Holling type III functional response when top predators are more effective at higher predator numbers and less effective at lower predator densities.The predator's persistence is maintained by utilizing the Holling type III functional response.
The main objective of the research is to investigate the dynamical behavior of the threespecies food chain model in the presence of various functional responses such as Crowley Martin functional response and Holling type III functional response.The local and global stability of the system are analyzed.The stability of the system depends on the presence of equilibrium points.The behavior of the model is examined using the Jacobian matrix.The system's overall stability and longevity are also evaluated using the Routh Hurwitz Criteria and Lyapunov function.By considering the above aspects, the mathematical model can be formulated which is given below: where  3 (0),  2 (0),  1 (0) > 0. The parameters of the model ,  1 , ,  After non-dimensionalization, the above system is of the form

Positivity and Boundedness
The Existence of positivity in the system with its initial condition guarantees the model.The following illustrates the system's positivity and boundedness:

Proof:
The system of Eq. 2 can be written in the following form with its initial condition as Published Online First: August, 2024 https://doi.org/10.21123/bsj.2024.10018P-ISSN: 2078-8665 -E-ISSN: 2411-7986 Baghdad Science Journal Integration of the above system of Eq. 3 now results in Thus all of the system of the Eq.2 solution remains positive in  + 3 .

Boundedness
The system's of Eq. 2 possible solutions are uniformly bounded in  + 3 .

Proof:
Since . Hence the theorem.

Equilibrium Points:
The stability of the system depends on the equilibrium points existing.The equilibrium points can be determined for the above system of Eq. 2.

Stability Analysis:
The Stability Analysis for the System of Eq. 2 is determined using the Jacobian matrix along with the existing equilibrium points.The Jacobian Matrix is of the form J( ] , 4 where All potential equilibrium points are employed in Eq. 4 to determine the stability of the model.The procedures for determining the stability are as follows. At the point  0 (0,0,0) in Eq. 4 then the matrix is given by J( 0 ) = [ At the Point  1 (1,0,0) in Eq. 4 then the matrix is . Thus the characteristic equation of the above matrix is The corresponding eigenvalues are −  1 < 0,  3 = − 2 < 0 .The system is locally asymptotically stable only if At the point  2 ( ̂1,  ̂2, 0) in Eq. 4 then the Matrix J( Thus the characteristic equation of the above matrix is given as follows The corresponding eigenvalues are  1 ,  2 and  3 .

Permanence
The Average Lyapunov function is used to demonstrate the system's of Eq. 2 permanence (Gard and Hallam) 22 .

Theorem:
The system of Eq. 2 is said to be permanent when it satisfies the following conditions as a)

Results and Discussion
Analytical findings are justified by the numerical simulation.Here the dynamical behavior of the three species model has been studied analytically with mixed functional response such as Crowley martin functional response and Holling type III functional response.The numerical simulations are done for the stability of the system.The Figs. 1-6 shows the stable, oscillatory behavior and phase portrait of the model.A numerical simulation has been done with the following set of parameters to show the dynamic behavior of the system of Eq. 2. The system's of Eq. 2 phase portraits are obtained, together with the associated time series graph.In the ecology the parameters value cannot be predicted exactly it varies.Therefore, the values are taken randomly for the simulation.

Conclusion
The mathematical framework of three species models in the ecosystem with the densities of Prey (N 1 ), Predator (N 2 ) and Top Predator (N 3 ) has been studied in this present paper.The interaction between these species with a mixed functional response is examined.The system's positivity and boundedness are studied.The system's feasible equilibrium points are all determined.The implicit premise in deterministic scenarios is the models which are created have been justified by their stability around the interior equilibrium.The local stability is examined using Routh-Hurwitz criteria and global stability by the Lyapunov function around the interior equilibrium point  3 ( 1 * ,  2 * ,  3 * ).Further, the conditions for permanence are analyzed.Computational simulations are done using MATLAB software.The paper can be further extended by adding any kind of delay in the model or other functional responses such as debeddington functional response, non-monotone functional response, and Holling type IV functional response.
https://doi.org/10.21123/bsj.2024.10018P-ISSN: 2078-8665 -E-ISSN: 2411-7986 Baghdad Science Journal Mathematical Framework Before the description of the mathematical model, some of the aspects are introduced.Three species models, comprising prey, predator, and top predator, have been considered.The three species are organized in a linear food chain, where the Predator hunts the Prey and the Top Predator hunts the Predator, as shown in a diagram.The feeding on the three species involves mixed functional responses such as Crowley Martin Functional Response between Predator and Prey while Holling type -III functional response between Top predator and Predator.In the absence of a Predator, the Prey population grows logistically with the intrinsic growth rate (r) and the carrying capacity (K).The Population densities of Prey, Predator and Top Predator over time are represented as  1 (),  2 () and  3 (t).The Crowley-Martin response function is affected by predator density, catch rate, handling time, and the level of disturbance among predators.The Crowley-Martin response functional response suggests that reciprocal interferences among predators still have a significant impact on eating rate when the prey population is huge.The Holling type-III functional response is defined by a sigmoidal relationship, where a substantial portion of predator devoured by the top predator increases in a density-dependent manner within specific predator population ranges.This physiological response enables the predator to persist.
Eigenvalues are  1 = 1 > 0,  2 = − 1 < 0,  3 = − 2 < 0. The system is unstable because the eigenvalues are real distinct and the point is saddle since one of its eigenvalue is an absolute value.

Figure 1 .
Figure 1.Variations of n 1 , n 2 , and n 3 along time t converge to an equilibrium state.

Figure 6 .
Figure 6.The system is asymptotically stable at the point E (0.8415, 0.676, and 1.322) . Numerous researchers have established different kinds of functional responses like Crowley Martin functional response, Beddington functional response, and Holling type I, Published Online First: August, 2024 https://doi.org/10.21123/bsj.2024.10018P-ISSN: 2078-8665 -E-ISSN: 2411-7986 1 ,  2 is the mortality rate of predator and Top Predator.The following method (nondimensionalization) is used to reduce the number of parameters in the system of Eq. 1.The dimensionless parameters are  1 = 1,  1 ,  2 and M are assumed to be positive.Here α, β are the predation rates of Predator and Top predator while  1 being the rate of transition from Prey to Predator and  1 the rate of transition from Predator to Top Predator.M is the half-saturation constant.