UNDERSTANDING THE INFLUENCE OF PREY ODOUR IN PREDATOR SPECIES: A THREE-SPECIES FOOD CHAIN STUDY

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INTRODUCTION
The preservation of ecological balance within the ecosystem has become a major challenge for humanity throughout time. Consequently, the significance of investigating a dynamic system, particularly an ecological system, is on the rise. An ecological system is a conceptual representation of a trophic chain that depicts the interrelated relationships between predators and prey. In the discipline of ecology, food chains are comprised of a sequential succession of species that function to supply sustenance to the species located in close proximity to them. The analysis of a predator-prey model in mathematics involves the development of a mathematical model aimed at addressing fundamental ecological issues related to food chains. The mathematical model proposed by Lotka [1] and Volterra [2], which was first introduced in the literature, has undergone subsequent modifications to effectively represent the dynamics of predator-prey populations. The aforementioned alterations encompass the integration of pragmatic elements such as sophisticated functional responses [3,4], time delay [5,6], stage structure [7,8], hunting cooperation [9,10], and fear effect [11,12]. The present study examines a food chain model comprising three distinct species, namely the prey, middle predator, and top predator.
The functional response plays a crucial role in the prey-predator model by defining the nature of the interaction between the predator and prey. Therefore, the integration of appropriate functional responses into the prey-predator model formulation is a critical component of the modelling procedure. Numerous scholars have undertaken investigations into a range of predatorprey models, considering diverse categories of functional responses, including but not limited to Holling type I-IV, Beddington-De-Angelis type, Crowley-Martin type, and Ratio-dependent.
The research conducted by Bhattacharjee et al. [13] pertained to an investigation of a model featuring two predators and one prey, utilising a Holling type I functional response. Majumdar et al. [14] conducted a study on a model of prey-predator interaction that incorporates a Holling type-II functional response. The utilisation of the Holling type III functional response has been observed in a model consisting of two predators and one prey, as presented by Didiharyono [15].
Numerous studies have been conducted in scientific literature pertaining to various predatorprey interactions to date. The primary aspect of predator-prey dynamics that forms the basis of their interaction is the mechanism through which the predator and prey identify each other, even prior to any actual encounter. One of the noteworthy and significant interactions between predators and prey pertains to the response of predators to the odour of their prey. It is widely acknowledged [20] that individuals generate unique odours due to various factors such as metabolic activities, hormonal changes, etc. . In most cases, such odours are inadvertently released into the environment [20,21]. The sense of odour can facilitate the differentiation and recognition of conspecifics and heterospecifics across various mammalian species. For instance, urinary scent marks are employed by mice as a means of communication with other members of their species in various social situations [21][22][23]. It has been established [20] that predators are attracted to the odours released by their prey. For example, blue crabs (Callinectes sapidus) have been observed to track prey odour plumes during their foraging activities [24]. According to studies [25], predators use informative cues like odours to efficiently locate and pursue potential prey. For instance, insectivorous avian species have the ability to utilise the chemical signals released by female moths, known as pheromones, to lure male moths as a means of detecting and capturing prey [26]. Canis lupus, commonly known as wolves, utilise olfactory cues to facilitate their hunting behaviour [27]. Also, the accumulation of odours resulting from roosting behaviour has the potential to attract predators and subsequently elevate the risk of predation [28]. Foxes exhibit an attraction to olfactory cues emitted by their prey [29]. Hence, the significance of odour is notably crucial. While the influence of prey odour on the process of predation is widely recognised in scholarly literature, there has been limited investigation into this subject matter within the context of a food chain. The impact of shelter on prey population in the presence of predator odour disturbance was examined by Shen et al. [30]. Also, Xu et al. [31] presented a model of predator-prey dynamics that incorporates the effects of odour disturbance and group defence. To the best of the authors' knowledge, these two articles are the sole works that examine the impact of odour that is relevant to our paper. So, there is a lack of literature in this area. To address this specific issue, the present work examines a food chain model comprising of three species while taking into consideration the impact of prey odour on the food chain. The study incorporates a linear function response to model the consumption of prey by the middle predator, and a Holling type II functional response to model the consumption of middle predator, by the top predator.
The present manuscript is structured in the following manner: The fundamental description of the mathematical model is presented in Section 3. Section 4 of the paper examines the boundedness of solutions and the presence of a positive equilibrium point. In Section 5, the discussion pertaining to the existence of equilibrium points and their local stability has been undertaken. The global stability of the equilibrium points is addressed in Section 6. Bifurcation analysis is done in Section 7. In Section 8, the dynamic behaviour of the considered model has been discussed through graphical representations of its numerical findings. Section 9 concludes this paper by summarising its key findings and contributions.

RESEARCH GAP AND CONTRIBUTION
Following a concise review of literature pertaining to food chain models in ecology, it is noted that a majority of researchers focused on various different effects such as fear effect, prey refuge, hunting cooperation by predators, etc that serve as extensions of the Lokta-Volterra model.
Nonetheless, the extant literature exhibits certain absences, which are itemised as follows: (a) The literature currently lacks sufficient investigation into the influence of odour on a threespecies model consisting of a solitary prey and two predators.
(b) Several studies [30,31] have explored the adverse impact of predator odour on prey, yet to date, no research has examined the potential positive effects of prey odour on predator populations, as far as the authors are aware.
In order to address the aforementioned gaps, our proposed three-species food chain model incorporates odour effect. In this paper, we have presented the olfactory impact as a detrimental factor for the prey and a beneficial factor for the predator. We consider prey odour aids predators in capturing their prey. Utilising this depiction, comprehensive theoretical and numerical evaluations pertaining to the stability and bifurcation of equilibrium points within the model have been conducted. Figure (1) depicts the research methodology utilised in the proposed model.

MATHEMATICAL MODELLING
In this section, a mathematical model is formulated to illustrate the influence of odour on a food chain system comprising three species. A system of three ordinary differential equations is utilised to depict the population dynamics of the prey, middle predator, and top predator, as well as their predation interactions. The mathematical formulation of the aforementioned biosystem commences with the conventional three-species food chain structure, followed by a gradual integration of the impact of prey odour on the intermediate predator (middle predator) within the food chain structure. The general form of a classical three-species food chain system is as follows.
The respective population sizes of the prey, middle predator, and top predator at a given time t are represented by the notations s(t), p 1 (t), and p 2 (t). The variable b is utilised to denote the birth rate of the prey, whereas the intra-species competition coefficient of the prey is represented by c. The mortalities that occur naturally within the populations of the prey, middle predator, and top predator are denoted by d, d 1 , and d 2 , respectively. The respective conversion efficiencies of the middle predator and top predator are denoted as a 1 and a 2 . Additionally, it can be noted that f 1 (s) and f 2 (p 1 ) represent the functional responses of the middle predator and the apex predator, respectively, in relation to their prey.
The release of olfactory cues by prey can play a pivotal role in facilitating the predatory behaviour of predators, thereby assisting in the identification and capture of their prey [32]. For instance, coyote (Canis latrans) [32] and bears [33,34], etc. exhibit a highly developed olfactory system that allows them to perceive potential food from a distance, even in the absence of visual stimuli. This enhances their ability to effectively locate and pursue their target. Following these instances, it is clear that the role of prey odour in predation by the predators within a food chain is of significant importance, neglecting this factor in mathematical modelling may result in a less impactful model. So, in this study, the role of prey odour in facilitating efficient tracking and hunting by predators is taken into consideration. We hypothesise that prey odour aids in the predation process of the middle predator. We also assume that the quantity of odour released by prey has a direct correlation with prey population size. To account for this relationship, we incorporate the expression (1 + γ s) into the functional response of the middle predator with respect to its prey. Consequently, the model (1) incorporating the odour effect becomes with initial conditions: s(0) = s 0 > 0 , p 1 (0) = p 0 1 > 0 , and p 2 (0) = p 0 2 > 0.
Here, the population sizes of the prey, middle predator, and top predator at a specific time t are denoted as s(t), p 1 (t), and p 2 (t) respectively.
Boundedness: The population of prey is consistently restricted by an upper bound as ds dt ≤ r 1 s(1 − s) which after integrating gives lim t→∞ s ≤ 1 . Now, we define x = s + p 1 + p 2 , and ν > 0 be an arbitary real number. Taking derivative w.r.t time 't' and using system (2), we get Now, we choose ν ≤ min(d 1 , d 2 ) and W is some constant, then . Now, we employ conventional results on differential inequalities and taking V is some constant Hence, it can be deduced that there is a positive value V, which solely relies on the parameters of system (2), such that 0 < w(t) ≤ V for values of t that are considerably large. Therefore, it can be concluded that all populations within the system are ultimately limited by upper bounds.
The existence conditions of these equilibrium points are as follows: (1) E 0 and E 1 always exists.

Local stability.
This section pertains to the examination of the local stability of the system (2) with the utilisation of the eigenvalue analysis approach, with regard to all equilibrium points. In order to assess the local stability of the equilibrium points of system (2), it is necessary to derive the Jacobian matrix at each equilibrium point. The Jacobian matrix at a point (s, p 1 , p 2 ) is given by  The following theorem pertains to the local stability of the equilibria of system (2).

Theorem 1.
(1) The population free equilibrium point E 0 is unstable.
The meanings of the symbols K i are given within the proof.
(4) The interior equilibrium point E is asymptotically stable if L i > 0, i=1, 2, 3 and L 1 L 2 > L 3 . The meanings of the symbols L i are given within the proof. Proof.
(1) The eigenvalues of the Jacobian matrix of the system (2) at E 0 are r 1 , -d 1 and -d 2 . As, the eigenvalues are of opposite signs which indicates E 0 , i.e., the population free equilibrium point is a saddle point, i.e., unstable.
(2) The Jacobian matrix of the system (2) at E 1 is . As a result, E 1 is locally stable if these conditions hold.
(3) For the equilibrium point E 2 , the Jacobian matrix is Here, Let us consider, δ 3 1 +K 1 δ 2 1 +K 2 δ 1 +K 3 = 0, be the characteristic equation of J (A 1 ,B 1 ,0) . Here, K 1 = −(p 11 + p 33 ), K 2 = −(p 12 p 21 − p 11 p 33 ), and K 3 = p 12 p 21 p 33 . By Routh-Hurwitz criterion, the top predator free equilibrium is locally asymptotically stable pro- (4) The variational matrix at the coexistent equilibrium point E is given by Here, where, Let, the characteristic equation of J (s ,p 1 ,p 2 ) is given by Here, L 1 = −(q 11 + q 22 ), L 2 = (−q 12 q 21 + q 11 q 22 − q 23 q 32 ), and L 3 = q 11 q 23 q 32 . Using Routh-Hurwitz criterion, the coexistent equilibrium point E is locally asymptotically  Proof. The Jacobian matrix of the system (2) at the equilibrium point E 1 is J (1,0,0) which is given by Now, the equilibrium point E 1 is asymptotically stable if −d 1 + (1 + γ)r 2 r 3 < 0 as stated in the previous section. From system (2), we have As per the comparison theorem, ally, it can be observed that p 2 (t) approaches to zero as p 1 (t) approaches to zero. Drawing from the theoretical framework of asymptotical autonomous systems as outlined in [35], system (2) can be simplified to a limiting system This suggests that the function s(t) approaches to 1. This implies that the equilibrium point E 1 exhibits global attractivity. Consequently, it can be concluded that the system denoted by E 1 exhibits global asymptotic stability.
Proof. The establishment of the global stability of the interior equilibrium E can be achieved through the construction of a Lyapunov function W, which is constructed in the following man- Hence, the time derivative of the aforementioned equation is dt . Using equations of system (2), we get where, κ = (p 1 − p 1 )( . Upon performing some calculations to simplify W 1 , W 2 , and W 3 , we get Now, we may assume θ 1 ≤ s ≤ θ 2 , ϑ 1 ≤ p 1 ≤ ϑ 2 , and Θ 1 ≤ p 2 ≤ Θ 2 and thus we get The abovementioned inequality (6) holds if the following criteria are met: Therefore, it can be concluded that dW dt < 0, i.e., the interior equilibrium is globally stable provided that the conditions (a), (b), (c), (d), and (e) are satisfied.

BIFURCATION ANALYSIS
The analysis of bifurcation is of paramount importance in comprehending intricate systems and forecasting their performance under varying circumstances. The study of bifurcations can offer valuable insights into the manner in which minor alterations in parameters can result in substantial alterations in the behaviour of a system.

Transcritical bifurcation.
The transcritical bifurcation will be discussed in this section.
The phenomenon of transcritical bifurcation is characterised by a significant alteration in the qualitative dynamics of a system, which occurs as a result of the exchange of stability properties between equilibrium points. Theorem 4. The system (2) yields a transcritical bifurcation at the critical parameter value γ = d 1 −r 2 r 3 r 2 r 3 = γ tc around the equilibrium point E 1 provided r 2 r 3 = 0.
Proof. The Jacobian matrix of the system (2) at the coexisting equilibrium point as previously stated in the preceding section. Taking γ = d 1 −r 2 r 3 r 2 r 3 = γ tc , we get Now, let us consider two eigenvectors U and V , which correspond to the zero eigenvalue of J (s ,p 1 ,p 2 ) γ=γ tc and J T (s ,p 1 ,p 2 ) γ=γ tc , respectively. After some computation, we get [36] is employed to establish the existence of a transcritical bifurcation on the parametric surface −d 1 + (1 + γ)r 2 r 3 =0 in the vicinity of E 1 . The prerequisites for transcritical bifurcation as per Sotomayor's theorem are outlined as follows: , and Thus, the verification of a transcritical bifurcation in the vicinity of E 1 at γ = d 1 −r 2 r 3 r 2 r 3 = γ tc is established utilising Sotomayor's theorem [36]. Alternatively, other parameters may also be utilised as bifurcating parameters.

Hopf bifurcation.
This subsection discusses the Hopf bifurcation. A Hopf bifurcation occurs when a system undergoes a significant change in stability, leading to the emergence of a periodic solution, at a specific critical value of a parameter. In this section, the Hopf bifurcation is examined in an analytical manner by investigating the coexistence equilibrium E with regard to the parameter denoting the odour effect γ, while holding all other parameters constant. In the theorem presented below, we demonstrate the existence of a Hopf bifurcation by considering the parameter γ as the bifurcation parameter. Proof. The characteristic equation of the system (2) in the vicinity of coexisting equilibrium E is given by the equation (4). When the parameter γ attains its critical value γ H , i.e., γ = γ H , then the equation (4) becomes The roots of the equation (7) are δ 1 = i √ L 2 , δ 2 = −i √ L 2 , and δ 3 = −L 1 . Now, in order to show the occurrence of Hopf bifurcation at γ = γ H , it is necessary to fulfil the transversality condition Let us consider, the roots are of the form Substituting the values of δ i (γ), i=1, 2 in equation (7) and calculating the derivatives, we get where, Since, χ 1 (γ H ) = 0, and χ 2 (γ H ) = L 2 (γ H ), so we have Consequently, the conditions of transversality are satisfied. This suggests that a Hopf bifurcation takes place when γ = γ H . Therefore, the theorem holds true. Other parameters can also be considered as bifurcating parameters.

NUMERICAL SIMULATIONS
The validation of analytical studies necessitates the numerical verification of the obtained outcomes. In this section, we explore the dynamics of system (2)     Additionally, we quantitatively verify Hopf bifurcations' presence, directional change, and stability. It is found that  In a similar way, we compute   (11d)). The existence of the aforementioned Hopf bifurcations is numerically validated through the utilisation of theorem (5). We find, the values the verification of the existence of Hopf bifurcation at γ = 2.114911 = γ H 1 has been established in accordance with theorem (5). In a similar vein, it is observed that The occurrence of a Hopf bifurcation is apparent at the parameter value r 2 = 0.032321 = r H 1 2 , as depicted in figure (12). In terms of numerical values, it can be observed that L 1 (r H 1 2 ) = 0.579743 > 0, L 2 (r H 1 2 ) = 0.31138 > 0, L 3 (r H 1 2 ) = 0.180643 > 0, ,   These conditions establish the supercritical nature of the Hopf bifurcation being discussed, in line with theorem (6). At the critical value of the parameter r 2 = 0.469544 = r tb 3 2 , an additional transcritical bifurcation arises, resulting in the instability of the interior equilibrium and the stability of the boundary equilibrium point. 8.5. Effects of growth rate of prey r 1 and death rates of the two predator species d 1 and d 2 on the food chain. The intrinsic growth rate of prey denotes the theoretical maximum rate at which a population of prey can expand in optimal circumstances, without any constraining factors such as resource scarcity, competition, or predation. The parameter r 1 in the biosystem (2) is associated with the intrinsic growth rate of the prey. By keeping all parameter values constant as specified in table (2), with the exception of r 1 , and manipulating r 1 as a variable, we have generated figure (15a). A transcritical bifurcation occurs when the parameter value r 1 reaches its critical value of r tb 1 = 0.341143. Consequently, the stability of the interior equilibrium is achieved, although the occurrence of a supercritical Hopf bifurcation at the critical value of r 1 = 0.923231 = r H 1 , destabilises the interior equilibrium point. This observation is readily discernible from figure (15a).
The mortality rate is a crucial ecological metric that impacts various aspects of population dynamics, species relationships, community organisation, ecosystem operation, and conservation endeavours. The variables d 1 and d 2 represent the mortality rates of the middle predator and top predator, respectively, within the biosystem (2). The graphical representation depicted in figure (15b) was constructed utilising the parameter values explicated in table (2)

SUMMARY
The biological process of predator-prey relationship is a fundamental subject in the field of ecology. It is very much evident that prey and predators utilise various strategies to maximise their biomass and increase their chances of survival. One such strategy employed by predators is to utilise the olfactory cues of their prey in order to enhance their likelihood of successful predation. The objective of this manuscript is to investigate a mathematical framework that represents a food chain system involving prey (s), intermediate predator, i.e, middle predator (p 1 ), and apex predator, i.e, top predator (p 2 ). The consistent utilisation of prey odour is assumed to aid the middle predator population in predation. The inclusion of prey odour into this predator-prey population model enhances its realism. The model assumes that the middle predator exhibits a linear functional response in consuming its prey, while top predator also utilise a Holling type II functional response in hunting its own prey i.e, middle predator. The dynamics of model (2) have been subjected to theoretical analysis to derive insights pertaining to its long-term behaviour. Theoretical evidences have been provided regarding the positiveness and boundedness of the solutions of the system (2). Subsequently, various equilibrium points were identified and subjected to a local stability analysis. The global stability of both the axial equilibrium point and the interior equilibrium point has been examined. A comprehensive numerical analysis has been performed to examine the Hopf bifurcation phenomenon with respect to several parameters. These parameters include the intake rate of middle predators (r 2 ), the intrinsic growth rate of prey (r 1 ), the coefficient of odour effect (γ), and the death rates of both predator species (d 1 and d 2 ), as well as the handling time of middle predators (the top predator's prey) by the top predator during predation (b). The findings are thoroughly addressed, providing insights into the impact of odour effect on the interactions within the food chain system. Additionally, we conducted a numerical investigation into the effects of the remaining parameters and illustrated our findings through the use of figures, which offer enhanced visual representation and ease of comprehension for a broader audience. The findings indicate that the dynamics of this system can be highly diverse, contingent upon the parameter ranges. These dynamics encompass the attainment of a solitary co-existence equilibrium, the eradication of either the top predator or both predator populations, and the persistence of oscillations or periodic behaviour.
Upon examination of the model, it has been determined that the trivial equilibrium point functions as a saddle point, thereby precluding the possibility of the extinction of all three species.
Additionally, it has been observed that the axial equilibrium, boundary equilibrium, and interior equilibrium states exhibit asymptotic stability under certain parametric constraints, as depicted in figures (3a), (4), and (5a). Moreover, the presence of the top predator free equilibrium negates the stability of the equilibrium solely occupied by the prey. Under certain parametric conditions, the prey-only equilibrium and the interior equilibrium can attain global stability, as demonstrated in figures (3b) and (5b) respectively.
Our numerical investigation into the influence of the odor-related parameter has yielded noteworthy and significant findings, demonstrating the manner in which the odour effect impacts long-term population dynamics. The absence of the odour effect in the model (2) has been observed to render the coexistence of all three species unattainable, ultimately leading to the collapse of the food chain (refer to figures (9) and (10a)). As the level of odour influence intensifies, the middle predator's viability is established through a transcritical bifurcation, and with subsequent elevations, the sustainability of all three species is affirmed via an additional transcritical bifurcation. At a certain threshold of odour impact, a Hopf bifurcation occurs in the population dynamics of system (2), resulting in periodic oscillation. The system is subsequently stabilised at a specific parameter value through another Hopf bifurcation (refer to figures (9), (10), and (11)).
The handling time of middle predators by the top predator during predation has been noted to play a significant role in the biosystem (2). The coexistence of all three species can be achieved when handling time is negligible, as depicted in figures (6) and (7a). Additionally, periodic fluctuations in the population of the three species arise from two supercritical Hopf bifurcations that take place at distinct values of the parameter linked to handling time. Furthermore, it has been observed that a transcritical bifurcation transpires at a specific value of the parameter b, which results in an alteration of the stability of the coexistence equilibrium state (refer to figures (6), (7), and (8)). Analogously, the system (2) exhibits intriguing dynamics with respect to the intake rate (r 2 ), prey growth rate (r 1 ), and the death rates of both predators (d 1 and d 2 ). As the intake rate r 2 varies, periodic fluctuations in the population dynamics of the system (2) are observed due to the existence of two supercritical Hopf bifurcations. Also, two transcritical bifurcations occur at different intake rates, indicating that it plays a significant role in sustaining all three species in the food chain (refer to figures (12), (13), and (14)). The population stability within the system (2) is also influenced by the intrinsic growth rate of the prey (r 1 ). At a specific value of the growth rate parameter of prey r 1 , a Hopf bifurcation is observed in the biosystem, leading to the emergence of periodic oscillations. Additionally, the intrinsic growth rate of the prey causes a transcritical bifurcation to occur (see figure (15a)).The population dynamics in the biosystem (2) are impacted by the mortality rates of both predators, denoted as d 1 and d 2 respectively. Periodic variations resulting from Hopf bifurcations and changes in the stability of the interior equilibrium due to transcritical bifurcations can be detected across various values of the parameters related to the death rates of the two predator species, namely d 1 and d 2 (see figure (15)).
This research provides evidence to substantiate the perspective that the odour of prey can serve as a pivotal element in maintaining cohabitation within a model comprising of multiple species.
In future research, it would be of academic interest to broaden the realism of the system (2) by incorporating real-world authentic data to estimate the parameters.