Complex Dynamics of an Aquatic Tri-Trophic Food Chain System with Holling Type IV Functional Response

Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 43 Retrieval Number:100.1/ijam.B121805021025 DOI: 10.54105/ijam.B1218.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Complex Dynamics of an Aquatic Tri-Trophic Food Chain System with Holling Type IV Functional Response Partha Ghosh Abstract: Complex dynamics of a modified Hastings-Powell (HP) model (phytoplankton-zooplankton-fish) with Holling type IV functional response are investigated in this article. Boundedness of the system has been established. A detailed study of the boundary equilibrium points and their local stability has been carried out. The condition of uniform persistence of the system has also been derived. Dynamical complexities and subsequent changes in the states of the system have been portrayed using numerical simulation. Modified HP model with Holling type IV functional response gives rise to a similar type of chaotic dynamics (inverted 'teacup attractor’) as observed in the original HP model with Holling type II functional response. Chaotic or stable dynamics are also numerically verified using Sil'nikov eigenvalue analysis. Keywords: Holling type IV, Boundedness, Uniform persistence, Chaos. Mathematics Subject Classification 2010: 92 XX Nomenclature: HP: Hastings-Powell I. INTRODUCTION Tri-trophic food chain systems with Holling-type functional response terms (type II and type III) have been investigated extensively over the years. Models with Holling type II functional response give rise to chaotic fluctuation in the system. Hastings and Powell [1] introduced a tri-trophic food chain model with Holling type II functional response term, which is widely accepted for explaining the chaotic movement of aquatic systems. There are many subsequent investigations of the type II system. Stollenwerk et. al. [2] analysed the Hollig type II system in both deterministic and stochastic environments. Myint and Wang [3] studied the Holling type II system with prey protection zone. In their pioneering work, Freedman and Waltman [4] derived the condition of persistence for these models. Holling type III systems are generally stable in nature. Francis et.al. [5] carried out a detailed stability analysis of a four-species Holling type III system. One can find many articles analysing different aspects of these systems. Manuscript received on 17 September 2025 | Revised Manuscript received on 29 September 2025 | Manuscript Accepted on 15 October 2025 | Manuscript published on 30 October 2025. *Correspondence Author(s) Dr. Partha Ghosh*, Department of Mathematics, Abhedananda, Mahavidyalaya, Sainthia, Birbhum, West Bengal, India. Email ID: [email protected], ORCID ID: 0009-0005-5386-5029 © The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/ Both Holling type II and Holling type III functional responses are monotonic in nature. However, many experimental and observational evidence found in natural ecosystems indicates the existence of a non-monotonic type response in community interactions. Also, dynamical complexities like bifurcation, chaos, etc., observed in these types of ecological systems can be explained more accurately with a non-monotonic type functional response than with traditional monotonic response terms. Andrews [6] first suggested a non-monotonic "Holling type IV" or "Monod-Haldane" functional response term as, 𝑓(𝑥)= 𝑝𝑥 𝑎+𝑏𝑥+𝑐𝑥2 A simplified "Monod-Haldane" function is of the following form, 𝑓(𝑥)= 𝑝𝑥 𝑎+𝑥2 Examples of non-monotonic type interactions can be found in many natural ecosystems, such as 'inhibition' in microbial dynamics, 'group defence' in population dynamics, and others. If the concentration of the nutrient reaches a certain high level, it may induce an inhibitory effect on the specific growth rate of the species. For example, one may investigate the process of waste decomposition or water purification by microorganisms. 'Monod-Haldane' functional response is like the Michaelis-Menten type function for low concentrations, but includes the inhibitory effect at high concentrations. Sarwardi et.al. [7] studied the effects of gestation delay and predator harvesting in the model system using non-monotonic functional response terms. Ghosh et.al. [8] investigated the bifurcation behaviour and stability of a three-species system with a non-monotonic functional response. Sokol and Howell proposed a simplified 'Monod-Haldane' function [9] during their experiments on the uptake of phenol by pure culture of Pseudomonas putida growing on phenol in continuous culture. They observed that the Holling type IV functional response fit their experimental data significantly better than other forms of functional response. In population dynamics, group defence [10] describes the phenomenon of decrease or prevention of predation due to the increased ability of the prey to defend better or disguise themselves when their numbers are large enough. Wolves can regularly attack a lone ox successfully. Small herds of Musk ox grazing together are attacked by wolves, but the success rate of the attacks is very low. On the other hand, successful attacks on larger herds are generally not observed. Daphnia can consume 'Filamentous algae' at low concentrations; However, they can survive the attacks at high concentration by Complex Dynamics of an Aquatic Tri-Trophic Food Chain System with Holling Type IV Functional Response 44 Retrieval Number:100.1/ijam.B121805021025 DOI: 10.54105/ijam.B1218.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. jamming their filtering apparatus [11]. In general, it can be said that when the per capita rate of predation decreases at sufficiently high prey densities, the Holling type IV functional response can be used with good effect. Examples of Holling type IV functional response can also be found in many other aspects of ecological study, as follows. Liu and Huang [12] analyzed a Holling type IV system with optimal harvesting and proved that maximum sustainable total yield and optimal economic profit do not exist with independent harvesting mode. Chai et al. [13] studied a stochastic Holling type IV system and provided sufficient conditions for persistence and extinction of populations. Alessandro et.al. [14] investigated a predator-prey system with generalised Holling type IV functional response and studied the interplay between the functional response and both strong and weak 'Allee effects' in the prey population. In the present article, a deterministic tri-trophic food chain system with a simplified 'Holling type IV' functional response term has been investigated. The present model may be considered to represent an aquatic plankton food chain system comprising phytoplankton (Coscinodiscus), zooplankton (Daphnia), and fish (salmon) populations. The aim is to study some critical properties like stability, boundedness, permanence, etc. Stability, especially global stability of the system indicates that the system is ecologically well organized and robust in nature. Boundedness of the system implies that the system is wellbehaved and the population biomass does not explode at any future time. Geometrically speaking, the solutions will always remain in the positive quadrant. From an ecological point of view, uniform persistence or permanence is a fundamental property of a model system, and several researchers have studied it [4]. Uniform persistence implies the survival of all populations in the system. In the case of uniformly persistent systems, strictly positive solutions are eventually uniformly bounded away from the boundary. The weak persistence property of a system only guarantees that the extinction of a particular species is not inevitable. But it may still happen that solutions approach arbitrarily close to the boundary for an extended period. In that case, any significant perturbation may drive the solution to the boundary, resulting in the extinction of the species. The idea of uniform persistence or permanence avoids this problem, and it is generally regarded as a more robust concept than weak persistence. One of the most crucial aspects of the dynamical complexities of a three-species system is the occurrence of chaotic oscillation. After the pioneering work of Hastings and Powell (HP) [1], chaos in model system has been studied by many researchers. The systems which are highly sensitive to initial conditions are prone to be chaotic. For these systems, a minute change in the initial conditions may cause a significantly diverging outcome. This behaviour of the system is defined as deterministic chaos or simply chaos. Chaotic systems do not have any definite pattern in their trajectories; instead, they always exhibit erratic oscillation. The chaotic behaviour of the present system has been investigated in detail through numerical simulation. The paper is organised as follows: In section II, a brief description of the model's formulation is provided. Boundedness of the system has been established in Section III. A detailed study of boundary equilibrium points and their local stability has been carried out in Section IV. The condition of uniform persistence of the system has been derived in Section V. Existence and uniqueness of the interior equilibrium point 𝐸∗(𝑥∗,𝑦∗,𝑧∗) have been investigated in section VI. A detailed numerical simulation is carried out in Section VII. A general discussion about the whole analysis is given in the concluding remarks in Section VIII. II. THE MATHEMATICAL MODEL The present model system (modified Hastings-Powell model [1] with Holling type IV functional response) describes a tri-trophic food chain model composed of a prey, a predator and a super predator whose population densities are denoted by 𝑋,𝑌 and 𝑍 respectively. Behaviours of the entire community are assumed to arise from the coupling of these interacting species, where 𝑍 preys on 𝑌 only and 𝑌 preys on 𝑋. The 'Holling type IV' or 'Monod-Haldane' functional response has been chosen as the predator response function for the species ( 𝑋,𝑌 ) and also for the species ( 𝑌,𝑍 ). With these assumptions, the following tri-trophic food chain model has been introduced, subject to the initial conditions. 𝑋,𝑌,𝑍>0. … (1) where 𝑅 is the intrinsic growth rate of the prey population and 𝐾 is its environmental carrying capacity. 𝐴1 and 𝐴2 are the maximal growth rates of the predator and the super predator, respectively. 𝐵1 and 𝐵2 are the half-saturation constants, 𝐶1−1 and 𝐶2 are conversion rates for prey to predator for the species 𝑌 and 𝑍 respectively. 𝐷1 and 𝐷2 These are the death rates of predators and super predators, respectively. All the parameters are assumed to be positive. To reduce the number of parameters and determine which combination controls the system's behaviour, it has been nondimensionalized in the following way. Let 𝑥=𝑋 𝐾,𝑦=𝐶1𝑌 𝐾,𝑧=𝐶1𝑍 𝐶2𝐾 and 𝑡=𝑅𝑇. Then system (1) takes the form (after simplification) … (2) where 𝑎1=𝐴1𝐾 𝑅𝐵1,𝑏1=𝐾2 𝐵1,𝑎2= 𝐶2𝐴2𝐾 𝐶1𝐵2𝑅,𝑏2=𝐾2 𝐵2𝐶12,𝑑1=𝐷1 𝑅 and 𝑑2=𝐷2 𝑅. Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 45 Retrieval Number:100.1/ijam.B121805021025 DOI: 10.54105/ijam.B1218.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. III. BOUNDEDNESS OF THE SYSTEM Boundedness of a system implies that the system is biologically well organized. In the following theorem, the boundedness of the system (2) is established. Theorem 3.1 All solutions of the system (2) which start in ℝ+ 3 are uniformly bounded. Proof. Let (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) be any solution of the system with favourable initial conditions. Since 𝑑𝑥 𝑑𝑡≤𝑥(1−𝑥), we have limsup 𝑥(𝑡)≤1. Let 𝑊=𝑥+𝑦+𝑧. Then 𝑑𝑊 𝑑𝑡 ≤𝑥−𝑑1𝑦−𝑑2𝑧 ≤2−𝛿𝑊, where 𝛿=min{1,𝑑1,𝑑2}. Therefore, 𝑑𝑊 𝑑𝑡 +𝛿𝑊≤2. So following Birkoff and Rota [15] we obtain, 0≤𝑊(𝑥,𝑦,𝑧)≤2𝛿+𝑊(𝑥(0),𝑦(0),𝑧(0)) 𝑒𝛿𝑡 For 𝑡→∞, we have 0≤𝑊≤2𝛿 Thus, all solutions of the system (2) remain in the following region. 𝐵ˆ={(𝑥,𝑦,𝑧):0≤𝑊≤2𝛿+𝜖, for any 𝜖>0} This proves the theorem. IV. BOUNDARY EQUILIBRIUM POINTS: EXISTENCE AND STABILITY The system (2) always has two boundary equilibrium points, namely 𝐸0(0,0,0) and 𝐸1(1,0,0). The planner equilibrium points are given by 𝐸2(𝑥ˆ,𝑦ˆ,0) and 𝐸3(𝑥′ ˆ,𝑦′ ˆ,0). A. Equilibrium Points 𝑬𝟎(𝟎,𝟎,𝟎) and 𝑬𝟏(𝟏,𝟎,𝟎) The variational matrices 𝑉(𝐸0) and 𝑉(𝐸1) are respectively given by, 𝑉(𝐸0)=[1 0 0 0 −𝑑10 0 0 −𝑑2], 𝑉(𝐸1) = [ −1 − 𝑎1 1+𝑏10 0𝑎1 1+𝑏1−𝑑10 0 0 −𝑑2 ] Two of the eigenvalues of 𝑉(𝐸0) are negative, and one is positive, so the point at the origin has both stable and unstable manifolds. For 𝑉(𝐸1), whenever 𝑎1 1+𝑏1>𝑑1, two of the eigenvalues are negative and one is positive, and as before, both stable and unstable manifolds exist for 𝐸1(1,0,0). Consequently both 𝐸0 and 𝐸1 become saddle points. B. Equilibrium Points 𝑬𝟐(𝒙ˆ,𝒚ˆ,𝟎) and 𝑬𝟑(𝒙′ ˆ,𝒚′ ˆ,𝟎) Since 𝑋≤𝐾 (where 𝐾 is the environmental carrying capacity of prey), it follows at once from the relation 𝑥=𝑋 𝐾 that 𝑥≤1. We can see that 𝑥 is given by the roots of the equation 𝑏1𝑑1𝑥2−𝑎1𝑥−𝑑1=0 and 𝑦 is given by, 𝑦=𝑥(1−𝑥) 𝑑1 From the expression of 𝑦 we can say that the existence of 𝑥 automatically implies the existence of 𝑦. The equilibrium points 𝐸2(𝑥ˆ,𝑦ˆ,0) and 𝐸3(𝑥ˆ′,𝑦ˆ′,0) exist if and only if 𝑎12> 4𝑏1𝑑12 and is given by, 𝑥ˆ=𝑎1+(𝑎12−4𝑏1𝑑12)1 2 2𝑏1𝑑1 𝑦ˆ=1 𝑑1(𝑎1+(𝑎12−4𝑏1𝑑12)1 2 2𝑏1𝑑1)(1−𝑎1+(𝑎12−4𝑏1𝑑12)1 2 2𝑏1𝑑1) and 𝑥′ ˆ=𝑎1−(𝑎12−4𝑏1𝑑12)1 2 2𝑏1𝑑1 𝑦′ ˆ=1 𝑑1(𝑎1−(𝑎12−4𝑏1𝑑12)1 2 2𝑏1𝑑1)(1−𝑎1−(𝑎12−4𝑏1𝑑12)1 2 2𝑏1𝑑1) So, we have the following result, Theorem 4.1 Equilibrium points 𝐸2(𝑥ˆ,𝑦ˆ,0) and 𝐸3(𝑥′ ˆ,𝑦′ ˆ,0) exist if and only if 𝑥<1 and 𝑎12>4𝑏1𝑑12. The variational matrix 𝑉(𝐸2) at 𝐸2(𝑥ˆ,𝑦ˆ,0) is given by, 𝑉(𝐸2)=[𝑣11 𝑣12 0 𝑣21 0 𝑣23 0 0 𝑣33] Where 𝑣11=1−2𝑥ˆ− 𝑎1𝑦ˆ 1+𝑏1𝑥ˆ2+2𝑎1𝑏1𝑥ˆ2𝑦ˆ (1+𝑏1𝑥ˆ2)2,𝑣12=− 𝑎1𝑥ˆ 1+𝑏1𝑥ˆ2 𝑣21=𝑎1𝑦ˆ 1+𝑏1𝑥ˆ2−2𝑎1𝑏1𝑥ˆ2𝑦ˆ (1+𝑏1𝑥ˆ2)2,𝑣23=− 𝑎2𝑦ˆ 1+𝑏2𝑦ˆ2,𝑣33 =𝑎2𝑦ˆ 1+𝑏2𝑦ˆ2−𝑑2 The characteristic equation of 𝑉(𝐸2) is given by, (𝜆2+𝐵𝜆+𝐶)(𝜆+𝑑2−𝑎2𝑦ˆ 1+𝑏2𝑦ˆ2)=0. Where 𝐵=−𝑣11 and 𝐶=−𝑣12𝑣21. The eigenvalues are given by, 𝜆1,2=−𝐵±(𝐵2−4𝐶)1 2 2,𝜆3= −𝑑2+𝑎2𝑦ˆ 1+𝑏2𝑦ˆ2. After a bit of calculation, it is found that 𝐶=−𝑣12𝑣21 is always negative. In contrast, cap 𝐵 is positive (or, negative) whenever 4𝑏1𝑑12 8𝑏1𝑑1−3𝑎1<𝑎1 1+𝑏1(or,4𝑏1𝑑12 8𝑏1𝑑1−3𝑎1>𝑎1 1+𝑏1) . Since 𝐶 is negative, the roots are always real but their sign will depend on the sign of 𝐵. If 𝐵>0 (i.e. 4𝑏1𝑑12 8𝑏1𝑑1−3𝑎1<𝑎1 1+𝑏1 ), then 𝜆1>0 and 𝜆2<0, which implies 𝐸2 becomes a saddle point in 𝑥𝑦 plane. If 𝐵<0 (i.e. 4𝑏1𝑑12 8𝑏1𝑑1−3𝑎1>𝑎1 1+𝑏1 ), then again 𝜆1>0 and 𝜆2<0, which implies 𝐸2 becomes a saddle point in 𝑥𝑦 plane. 𝐸2 becomes locally asymptotically stable in 𝑧 direction if and only if −𝑑2+𝑎2𝑦ˆ 1+𝑏2𝑦ˆ2<0. Therefore, we get the following theorem. Theorem 4.2 𝐸2 is a saddle point in the xy plane. 𝐸2 becomes locally Complex Dynamics of an Aquatic Tri-Trophic Food Chain System with Holling Type IV Functional Response 46 Retrieval Number:100.1/ijam.B121805021025 DOI: 10.54105/ijam.B1218.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. asymptotically stable (or unstable) in 𝑧 direction if and only if −𝑑2+𝑎2𝑦ˆ 1+𝑏2𝑦ˆ2<( or >)0. Proceeding similarly to above, we can obtain the following result for 𝐸3(𝑥ˆ′,𝑦ˆ′,0). Theorem 4.3 (a) 𝐸3 is locally asymptotically stable in the xy plane if and only if 4𝑏1𝑑12 8𝑏1𝑑1−3𝑎1>𝑎1 1+𝑏1 and unstable if 4𝑏1𝑑12 8𝑏1𝑑1−3𝑎1<𝑎1 1+𝑏1. (b) 𝐸3 is locally asymptotically stable (unstable) in the 𝑧 direction if and only if −𝑑2+𝑎2𝑦ˆ′ 1+𝑏2𝑦′ ˆ2<( or >)0. V. UNIFORM PERSISTENCE Uniform persistence or permanence is a stronger concept than weak persistence or persistence. Permanence of the present system is established in the following theorem: whenever there exists a finite number of limit cycles in 𝑥𝑦 Plane. Theorem 5.1 Let 𝑎1 1+𝑏1>𝑑1,4𝑏1𝑑12 8𝑏1𝑑1−3𝑎1>𝑎1 1+𝑏1 and there exists a finite number (say 𝑛 ) of periodic solutions 𝑥= 𝜙𝑖(𝑡),𝑦=𝜓𝑖(𝑡)(𝑖=1,2,…𝑛) in the 𝑥𝑦 plane. Then the system is uniformly persistent, provided for each periodic solution of period. 𝑇ˆ, −𝑑2+1 𝑇ˆ∫ 𝑇ˆ 0𝑎2𝜓𝑖(𝑡) 1+𝑏2𝜓𝑖2(𝑡)𝑑𝑡>0,(𝑖=1,2,…𝑛) Proof. Let 𝜃(𝑋ˆ) be the orbit through the point 𝑋ˆ=(𝑥,𝑦,𝑧) and Ω(𝑋ˆ) be the omega limit set of the orbit through 𝑋ˆ. We note that Ω(𝑋ˆ) is bounded. First, we shall show that 𝐸0 does not belong to Ω(𝑋ˆ). If 𝐸0∈Ω(𝑋ˆ), by the Butler-McGehee lemma [4], there exists a point 𝑃 in Ω(𝑋ˆ)⋂ 𝑊𝑠(𝐸0). Where 𝑊𝑠(𝐸0) the stable manifold of 𝐸0. Since 𝜃(𝑝) lies in Ω(𝑋ˆ) and 𝑊𝑠(𝐸0) is the 𝑦𝑧 plane, we found that 𝜃(𝑝) is unbounded, which is a contradiction. Next 𝐸1 does not belong to Ω(𝑋ˆ), for then since 𝐸1 is a saddle point from the condition 𝑎1 1+𝑏1>𝑑1, by the ButlerMcGehee lemma [4], there exists a point 𝑃 in Ω(𝑋ˆ)∩ 𝑊𝑠(𝐸1). 𝑊𝑠(𝐸1) is the 𝑥 axis implies that an unbounded orbit lies Ω(𝑋ˆ), which is a contradiction to the boundedness of the system. Since 𝐸2 is always a saddle point, proceeding as above, we conclude that 𝐸2 does not belong to Ω(𝑋ˆ). Now we only need to consider the equilibrium point 𝐸3. We shall show that no periodic orbit in the 𝑥𝑦 plane or 𝐸3 belongs to Ω(𝑋ˆ). According to condition (b) of the theorem, eigenvalues 𝜆1′,𝜆2 ′ of 𝑉(𝐸3) have a negative real part. Let 𝛾𝑖(𝑖=1,2,…,𝑛) denotes the closed orbit of the periodic solution ( 𝜙𝑖(𝑡),𝜓𝑖(𝑡) ) in 𝑥𝑦 plane such that 𝛾𝑖 lies inside 𝛾𝑖−1. The variational matrix 𝑉𝑖(𝜙𝑖(𝑡),𝜓𝑖(𝑡),0) corresponding to 𝛾𝑖 is given by, 𝑉(𝐸3)=[𝐹(𝜙𝑖(𝑡),𝜓𝑖(𝑡))𝜙𝑖(𝑡)𝐹𝑦(𝜙𝑖(𝑡),𝜓𝑖(𝑡))0 𝜓𝑖(𝑡)𝐺𝑥(𝜙𝑖(𝑡),0)𝐺(𝜙𝑖(𝑡),0)𝜓𝑖(𝑡)𝐺𝑧(𝜙𝑖(𝑡),0) 0 0 𝐻(𝜓𝑖(𝑡))] where, 𝐻(𝜓𝑖(𝑡))=−𝑑2+𝑎2𝜓𝑖(𝑡) 1+𝑏2𝜓𝑖2(𝑡) Computing the fundamental matrix of the linear periodic system 𝑀′=𝑉𝑖(𝑡)𝑀,𝑀(0)=𝐼 we find that its Floquet multiplier in the 𝑧 direction is 𝑒𝑖𝜂(𝑡). Then, following the approach of Kumar and Freedman [16], we conclude that no 𝛾𝑖 lies in Ω(𝑋ˆ). Thus Ω(𝑋ˆ) lies in the positive octant, and the system is persistent. Finally, since only the closed orbits and the equilibria form the omega limit set of the solutions on the boundary of ℝ+ 3, and the system (2) is dissipative, then by the main theorem in [17], the system (2) is uniformly persistent. VI. THE INTERIOR EQUILIBRIUM POINT: EXISTENCE AND UNIQUENESS In this section, the conditions for the existence and uniqueness of the interior equilibrium point (𝑥∗,𝑦∗,𝑧∗) of the system (2) has been derived. We can easily see that there exist two values of 𝑦∗. 𝑦1∗=𝑎2+(𝑎22−4𝑏2𝑑22)1 2 2𝑏2𝑑2,𝑦2∗=𝑎2−(𝑎22−4𝑏2𝑑22)1 2 2𝑏2𝑑2 If 𝑎22=4𝑏2𝑑22 then the two values are equal and given by 𝑦∗=𝑎2 2𝑏2𝑑2. 𝑥∗ is the positive solution of the equation 𝑥3−𝑥2+1 𝑏1𝑥+ 𝑎1𝑎2−2𝑏2𝑑2 2𝑏1𝑏2𝑑2=0and 𝑧∗ is given by, 𝑧∗=𝑥∗(1−𝑥∗)−𝑑1𝑦∗ 𝑑2 We note that 𝑥∗ is less than 1, and by Descartes' rule of signs, there always exists at least one positive root of the equation. Also by the relation between roots and coefficients of cubic equation we find that if (𝑎1𝑎2−2𝑏2𝑑2 2𝑏1𝑏2𝑑2+1 3𝑏1−2 27)2+ 4(27𝑏1−1 27𝑏12)3>0 then two of the roots of the equation become imaginary and consequently unique positive root exists for the equation. So, we have the following theorem. Theorem 6.1 The unique positive equilibrium of the system (2) exists if and only if 𝑎22=4𝑏2𝑑22 and (𝑎1𝑎2−2𝑏2𝑑2 2𝑏1𝑏2𝑑2+1 3𝑏1−2 27)2+4(27𝑏1−1 27𝑏12)3>0. VII. NUMERICAL SIMULATION A. Transition from Stable to Chaotic Dynamics The present system has been studied by varying the parameter 𝑏1 (half saturation constant) gradually keeping the values of the other five parameters fixed as taken in the original HP model [1]. 𝑎1=5,𝑎2=0.1,𝑏2=2,𝑑1= 0.4,𝑑2=.01. The initial values are taken as 𝑥(0)= 0.75,𝑦(0)=0.15,𝑧(0)=9.75. For these values of parameters and initial conditions, the interior equilibrium point 𝐸∗(𝑥∗,𝑦∗,𝑧∗) is given by, Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 47 Retrieval Number:100.1/ijam.B121805021025 DOI: 10.54105/ijam.B1218.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. 𝑥∗=0.782544,𝑦∗=0.10208, 𝑧∗=12.93363. It has been observed that for 𝑏1∈(0,1.5), the system (2) exhibits stable behaviour (see Fig. 1(a)). Now, as the value of 𝑏1 is gradually increased, the system transits through different phases and eventually turns chaotic. It has also been observed that dynamics is susceptible to small changes in system parameters, and system dynamics can change significantly within a very short interval. Limit cycle oscillation can be observed in the system for 𝑏1∈(1.58,1.7) (Fig.1(b)). Further period doubling occurs (Fig.1(c)) for 𝑏1= 1.78. Finally, chaotic oscillations appear in the system along with the evolution of the famous inverted teacup attractor for 𝑏1∈(1.83,3.2). Onset of chaotic fluctuation can be observed in Fig.1(d). As observed in the original HP model, trajectories start in the "handle" of the "teacup," move to the wide part, spiral along the teacup to the narrow end, enter the handle again, and continue with the same pattern. Also, all the trajectories in the "handle" of the "teacup" are very close together, which contributes to the sensitive dependence of future dynamics on the current state. Fig.2(a) shows the chaotic time series for 𝑏1=2.2 and Fig. 2(b) is the corresponding chaotic attractor (inverted teacup). When the value of 𝑏1 is increased further, the shape of the chaotic attractor changes, but the dynamics of the system remain chaotic. Also, no species extinction is observed throughout the interval. [Fig.1(a) Stable Phase Portrait for 𝒃𝟏=𝟏.𝟓] [Fig.1(b) Limit Cycle for 𝒃𝟏=𝟏.𝟔𝟖] [Fig.1(c) Period Doubling for 𝒃𝟏=𝟏.𝟕𝟖] [Fig.1(d) Onset of Chaotic Fluctuation for 𝒃𝟏=𝟐] [Fig.2(a) Chaotic Time Series for 𝒃𝟏=𝟐.𝟐] [Fig.2(b) Chaotic Attractor for 𝒃𝟏=𝟐.𝟐] B. Sil'Nikov Chaos If the equilibrium point of the system is a saddle focus, and the eigenvalues 𝛾‾ and 𝛼‾±𝑖𝛽‾ of the corresponding Jacobian matrix satisfies the following Sil'nikov inequality [18] Complex Dynamics of an Aquatic Tri-Trophic Food Chain System with Holling Type IV Functional Response 48 Retrieval Number:100.1/ijam.B121805021025 DOI: 10.54105/ijam.B1218.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. 𝛽‾≠0,𝛾‾𝛼‾<0, and |𝛾‾|>|𝛼‾|≧0 then the system may have Sil'nikov chaos in some neighbourhood of the equilibrium point. To establish the existence of chaotic oscillation in the system according to Sil'nikov criteria, the eigenvalues have been calculated for the specified set of system parameter values using MATD. It has been observed that Sil'nikov conditions are not satisfied in the region 𝑏1∈(0,1.65), but they are satisfied with 𝑏1≥1.65. For example, when 𝑏1= 1.2 (in the stable region) the eigenvalues are -0.0385 and −0.1623±𝑖0.4075. But for 𝑏1=2.2 (in the chaotic region) the eigenvalues are -0.6114 and 0.0651±𝑖0.797. There are several articles containing the analysis in this regard [19]. VIII. CONCLUSION Complex dynamics of a deterministic three-species food chain model with a simplified Holling type IV functional response is investigated in this article. As mentioned in the beginning, the present model system may be considered to represent an aquatic plankton food chain system comprising phytoplankton (Coscinodiscus), zooplankton (Daphnia), and fish (salmon) populations. More specifically speaking, it may be envisaged as the aquaculture of salmonids (salmon and steelhead). Aquaculture is the process of farming and harvesting salmonids under controlled conditions for both commercial and recreational purposes. The Atlantic salmon, Chinook salmon, rainbow trout and brown trout are the most commonly farmed salmonids for recreational and subsistence fishing in America and Europe. Stability of these types of aquacultures can be affected by various factors like 'uncontrolled harvesting', 'poaching', 'uncontrolled recreational and subsistence fishing' or 'escape into wild habitats during storms', etc. These external factors may drive the stable system to a chaotic one. In the present investigation, it has been found that the system is uniformly bounded and permanent. Existence of equilibrium points and their stability conditions have been derived. Interestingly, it has also been observed that the system exhibits chaotic oscillation within a specific range of system parameters. It is observed that the system transits through different states, from Stable to limit cycle behaviour, then to period doubling, and finally to the chaotic attractor. The existence of chaos in the system has also been established through Sil'nikov eigenvalue analysis. Although chaotic fluctuation is observed in the present system, it is interesting to note that no population extinction occurs. So, uniform persistence of the system is validated by the numerical findings. It should be noted that although chaos is predicted in many model systems, its example in the natural world, particularly in terrestrial ecosystems, is highly elusive. Also, uniformly persistent systems are not, in general, chaotic in nature. But this is not the case for aquatic systems, as they are more prone to chaos. The results obtained here through analytical and numerical simulation can be used as a tool to determine the tipping point between the persistence and extinction of a population. As a matter of fact, many ecological species are in a threatened condition and may be stabilised by suitable management. Further investigation is necessary for proper monitoring of the ecological system. DECLARATION STATEMENT Some of the references used in the present article are pretty old. For example, Reference no. [1], [4], [6] [9], [16], [17] and [18]. But readers will find that these articles are extremely important for the results developed in the present study. Most of them are actually pioneering works in their respective fields. They have also been cited extensively by researchers in several articles within the fields of mathematical biology and ecology. I must verify the accuracy of the following information as the article's author. ▪ Conflicts of Interest/ Competing Interests: Based on my understanding, this article has no conflicts of interest. ▪ Funding Support: This article has not been funded by any organizations or agencies. This independence ensures that the research is conducted with objectivity and without any external influence. ▪ Ethical Approval and Consent to Participate: The content of this article does not necessitate ethical approval or consent to participate with supporting documentation. ▪ Data Access Statement and Material Availability: The adequate resources of this article are publicly accessible. ▪ Author’s Contributions: The authorship of this article is contributed solely. REFERENCES 1. Hastings, A., Powell, T. (1991). Chaos in a Three-Species Food Chain. In Ecology (Vol 72, Issue 3, pp 896-903). DOI: https://doi.org/10.2307/1940591/ works, works remain significant, see declaration 2. Stollenwerk, N. Aguiar, M. Kooi, B.W.(2022). Modelling Holling type II Functional Response in Deterministic and Stochastic Food Chain Models with Mass Conversion. In Ecological Complexity (Vol 49, March 2022, 100982). DOI: https://doi.org/10.101/j.ecocom.2022.100982 3. Myint, A. Z. Wang, M.(2022). 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Butler, G. J. Waltman, P. Freedman, H. I (1986) Uniformly persistent systems. In Proceedings of the American Mathematical Society ( Vol. 96, pp 425-430). DOI: https://doi.org/10.1090/S0002-9939-19860822433-4, works remain significant, see declaration 18. Zhou, L. Chen, F. (2008). Sil'nikov chaos of the Liu system. In Chaos, (Vol 18, March 10, 013113). DOI: https://doi.org/10.1063/1.2839909, works remain significant, see declaration 19. Ding, Y. Zheng, L. (2023) Existence of homoclinic orbit of Shilnikov type and the Application in Rossler System. In Mathematics and Computers in Simulation (Vol 206, April 2023, pp 770-779). DOI: https://doi.org/10.1016/j.matcom.2022.12.013 AUTHOR’S PROFILE Dr. Partha Ghosh M.Sc. (Pure Mathematics), M.Phil., PhD, obtained the degree of M.Sc. and M.Phil from The University of Calcutta with M.Phil Dissertation work in the field of Fractal Geometry. He later received his PhD from the Indian Institute of Engineering Science and Technology (IIEST), Shibpur, in the field of Mathematical Biology. He is an Assistant Professor currently associated with the Department of Mathematics at Abhedananda Mahavidyalaya, Sinthia, Birbhum, West Bengal. His current area of research includes Dynamical Systems, Ecology, and Stochastic Analysis, among others. He is engaged in various research activities with fellow collaborators in college and other institutions in the area of Mathematical Biology, Mathematical Epidemiology, and noiseinduced ecological systems. Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of the Lattice Science Publication (LSP)/ journal and/ or the editor(s). 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