Spatiotemporal Tipping Induced by Turing Instability and Hopf Bifurcation in a Population Ecosystem Model with the Fear Factor

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Introduction
It is proposed by Leslie [1] that predator-prey is the basic component in ecological competition systems and the most fundamental species relationship in nature.Predation plays an important role in ecological population systems.Predation controls the density of biological populations and directly afects the amount of natural resources.In view of predation, there has been tremendous progress in the dynamics of predator and prey [2][3][4][5].Recently, many populations have been on the verge of extinction due to destructive human exploitation, pollution, and hunting that have had a huge impact on nature and species.Te studies of ecological competition dynamic models have been pushed to the hot spot [6][7][8][9].Predator-prey systems are an essential component to revealing the competition mechanism, growth law, and control strategy among populations, so it is important to establish a scientifc predator-prey model.
Local stability of the system means that if the initial state is adjacent to the equilibrium state, the system will not vibrate, and its state trajectory will eventually fall at the equilibrium state.Specially, Hopf bifurcation is a dynamic bifurcation phenomenon, which shows that when the system parameter varies near the critical value, the stability of the equilibrium point will change and periodic solutions will be generated.From a dynamical point of view, the quantitative relationship between prey and predators sometimes has periodic fuctuations, and the limit cycle may arise due to the Hopf bifurcation, which is an equal amplitude oscillation caused by the critical state.In this sense, the Hopf bifurcation is an important issue for the study of the dynamics of population competition models.Terefore, most of the previous work focused on the existence of coexisting equilibria and bifurcation as well as the dynamic behavior of nonlinear systems with periodic solutions and chaos (see [10][11][12][13][14][15]). However, these existing ecological competition models are characterized by ordinary diferential equations (ODEs) or fractional ordinary diferential equations (FODEs), which can only describe the evolution of species in the time dimension but cannot depict the migration in space.In the reaction-difusion system proposed by Turing [16,17], the spatial heterogeneity caused by the internal difusion characteristics of the system results in the loss of system symmetry and makes the system self-organization to produce some spatial patterns.Te process of pattern formation is called Turing instability (Turing bifurcation).Te symmetry of the system is broken, leading to the formation of Turing patterns.Terefore, we call this phenomenon Turing instability caused by difusive reaction [18].In fact, species often migrate from high-density areas to low-density areas in search of better survival conditions.Te uneven spatial distribution of species has an impact on predation behaviors.Hence, more and more scholars have concentrated on predator-prey models with difusion terms, which can be described by reaction-difusion equations.Te stability, difusion-induced Turing instability, Hopf/Bogdanov-Takens bifurcations, and Turing pattern have been studied by many scholars [19][20][21][22][23][24].
Te fear factor of predators also has a signifcant impact on interspecies predation.As an indirect efect, fear from predators can afect the population size of prey.Such fear can reduce reproductive capacity.An increase in the number of predators results in greater fear of prey, and then the growth rate of prey drops sharply.Te reduction in prey density decreases the efciency of predator feeding behavior.Terefore, the growth rate of the predator is declining [25,26].Terefore, the fear factor of prey towards predators is also essential for the dynamic model of ecological competition.On the contrary, processes such as the gestation of prey, digestion of predators, spread of diseases, and human impact are not instantaneous behaviors.Te introduction of time delays to predator-prey systems is more meaningful [27][28][29].Te fear of time delay can have a signifcant efect on the dynamics of ecological population systems.Tis article considers primarily the efect of fear delay on the dynamic behaviors of population competition systems.
It is worth noting that despite considerable advances in the dynamics of population-competitive ecosystems, there are still many issues that deserve further study.Little is known about the tipping phenomena of predator-prey systems.It is observed that a variety of real-world complex systems can exhibit abrupt transitions in dynamical patterns and states of behaviors, then they are called to tip from one emergent mode to another [30,31].Some such transitions that can infuence the production and lives of all human beings are the shutdown of the thermohaline circulation, global changes in climate, extinction of species in ecosystems, the crashing of fnancial markets, and the surge of COVID-19 epidemics [32][33][34][35][36][37].Tese tippings occur mostly due to small perturbations to the threshold values of state variables or system parameters, giving rise to dramatic qualitative changes in the dynamics.Tey have often had a tremendous infuence and are mostly irreversible.
Habitat transformations and connections between species can disappear in an ecological system.As the fraction of vanishing connections continues to increase beyond a threshold, the ecological system may arrive at a tipping point past which the whole system crumbles.Ten, all species die at the same time [38].Terefore, to understand the dynamical mechanism of the tipping point of predatorprey systems and to predict its occurrence are the overriding problems of broad interest and great importance.Research studies have shown that the primary deterministic mechanisms behind several of the tipping events are so-called bifurcations [31,39].Tis article aims to propose a population competition model having fear delays and difusion terms, and then predicts and exploring the tipping points induced by Turing instability and Hopf bifurcations.
We organize the rest of the article as follows.A difusive predator-prey model including fear factors is presented in Section 2 Section investigates Turing instability-induced tipping for the difusive population competition model without time delays.Te impact of difusion terms on the tipping dynamics is discussed.Meantime, both time delay and difusion factors are considered, and the tipping point induced by Hopf bifurcation is studied.Te Hopf bifurcation criterion of the population evolution model is proven.Section 4 determines the Hopf bifurcation direction in order to gain insight into tipping mechanisms.Section 5 provides several examples to testify to the validity of the theoretical analysis.Te conclusion is drawn in Section 6.

Model Formulation
We establish a difusive predator-prey system including fear delay in this section.
In [40], a classical predator-prey system is summarized by Te basic assumptions of system (1) are described below: (i) ρ and θ stand for the densities of prey and predator, respectively.K and r represent the carrying capacity and the intrinsic growth rate of the logistical growth of prey population.(ii) mρθ/(Aθ + ρ) describes a modifed ratio-dependent functional response of Holling type II.(iii) s stands for the inherent growth rate of predators.
Te carrying ca-pacity of predators is ρ/h, which is described by a function of prey population.(iv) All the coefcients K, r, m, A, h, s > 0.
Since the fear factor is essential for prey growth, the fear delay is naturally introduced for system (1) as follows: where C > 0 denotes the fear coefcient and τ′ is the fear delay.
Considering the impact of populations on diferent spatial locations and the heterogeneous distribution of food resources, we put forward a novel difusive predator-prey system, which can be described by with Neumann boundary and initial conditions where ρ (t, ϖ) and θ (t, ϖ) represent the population density of prey and predators at time t and spatial location ϖ, respectively.∆ is the Laplacian operator and the positive difusion coefcients  d 1 and  d 2 are related to ρ (t, ϖ) and θ (t, ϖ), respectively.Te bounded domain Θ � (0, ιπ) (ι > 0) has a smooth boundary zΘ.ε represents the outward unit normal vector on zΘ.ρ 0 (t, ϖ) and θ 0 (t, ϖ) are the nonnegative continuous functions.
Te difusive predator-prey system (3) can be conversed by the following transformation: and it becomes where Te dynamical behaviors of system (6) are equivalent to those of system (3).Te following derivation revolves around system (6).
Remark 1. Assuming that there is no difusion efects, system (6) may degenerate to the system described by ODEs.Te emergence of difusion terms does not change the equilibria.Now, we make the transformation of variables: We drop the hats for the sake of simplifed calculation and the linearized system of ( 6 where Let D � diag {d 1 , d 2 }.System (10) can be written as follows: in which Te characteristic equation of system ( 12) is as follows: As we know, the eigenvalues of Δ are −k 2 (k∈ {0, 1, 2, . ..}) on the phase space X and the corresponding eigenfunctions are given by where So, equation ( 14) can be turned into the following form: in which 3.1.Tipping Induced by Turing Instability.We intend to probe into the impact of difusion on the dynamical evolution of system ( 6) without time delays and study the tipping point caused by Turing instability in this subsection.When τ � 0, equation ( 17) becomes If d 1 � d 2 � 0 holds, equation ( 19) can be reduced to the following difusion-free form: where We make the following assumption: Lemma 1.We assume that d 1 � d 2 � 0 and τ � 0. If (H1) is satisfed, then the trajectories of system ( 6) converge to the equilibrium E * .
Te proof of Lemma 1 is straightforward on the basis of Routh-Hurwitz criterion.
Te following theorem is true.
Theorem 1 (Turing instability-induced tipping).Te following results are true for system (6) without time delay.
(i) If d 1 � d 2 � 0 and (H1) hold, the trajectories of system (6) without time delay converge to E * .(ii) If d 1 > 0 and d 2 > 0 and (H2) hold, there exists at least one k ∈N 0 such that system (6) without time delay becomes unstable at E * , while a Turing instability occurs.Hence, this results in a Turing instabilityinduced tipping.
Remark 2. It should be pointed out that this article adopts the linearization method [23,24,29,41] to deal with the dynamics analysis of systems, including the local stability, Turing instability and Hopf bifurcation.It is common knowledge that Lyapunov's second method is important to stability theory of dynamical systems and control theory.However, this method is not suitable for investigating the dynamics of the ecological competitive system with delay and difusion proposed in this article.Te Lyapunov stability criterion can only give a sufcient condition for the stability 4 Complexity of a system.In this article, not only the condition of the local stability is established but also the boundary of stability (the onset of Hopf bifurcation) is determined.
Remark 3. It is obvious from Teorem 1 that the predatorprey system (6) without spatial difusion terms is stable at the endemic equilibrium, but the introduction of difusions makes the system unstable.Tis phenomenon is called Turing instability.Meanwhile, a transition from a stable mode to an unstable one is called a tipping caused by Turing instability.

Tipping Induced by Hopf
Bifurcation.Te impacts of difusion and time delays on the spatiotemporal evolution of system (6) forecast the tipping point induced by Hopf bifurcation in this section.Equation ( 19) is the characteristic equation for system (6) without delay.We can give the stability condition of E * of system (6) as follows: Lemma .Assume that τ � 0 and d 1 > 0, d 2 > 0. If (H3) is true, the trajectories of system ( 6) converge to the equilibrium E * for any k ∈ N.
Proof.Note that When (H3) is satisfed, all eigenvalues of equation ( 19) are located in the left half plane.Tis means that the trajectories converge to E * for any k∈ N.
Theorem 2. Suppose (H3) holds.We come to the following statements.
Remark 4. Teorem 2 illustrates that, as the bifurcation parameter-delay passes through the threshold τ 0 that is the tipping point, system (6) goes from the stable mode to the unstable state at E * and a periodic oscillation occurs.Such a transition is called the tipping driven by Hopf bifurcation.
It is worth noting that the tipping can be predicted by the expression of the tipping point τ 0 .
We change the difusion coefcients to  d 1 � 0.0088 and  d 2 � 0.04 and other parameters are consistent with those in Figure 1.Assumption (H2) is not satisfed, and the trajectories of system (3) still tend to E * as displayed in Figure 2. Furthermore, we set  d 1 � 0.0088 and  d 2 � 0.288.It can be verifed that (H2) is reached.From Teorem 1, E * begins to oscillate in the spatial region and the stripe patterns appear, which are depicted in Figures 3 and 4, respectively.Tis implies that system (3) transitions into an unstable state and the Turing instability-induced tipping occurs.
In addition, we set r � 0.8, m � 0.8, 6, and l � 1.We can compute two critical values, 4.0855 and 11.7601, which are two tipping points.When τ′ < 4.0855, the orbits converge to equilibrium E * as illustrated in Figure 8, and when τ′ crosses the tipping point of 4.0855, the orbits turn away from E * and the Hopf bifurcation emerges as displayed in Figure 9. Furthermore, when τ′ continues to increase and passes through the other tipping point 11.7601, E * returns to the stability as depicted in Figure 10.It is found that the spatiotemporal evolution of system (3) switches between stable focus and limit cycles several times with the increase in time delay.Tis reveals that there may be many tipping points in ecological competition systems, and the tipping may occur many times as the fear delay increases.
Finally, we explore the infuence of fear factors on the densities of species.We fx r � 0.8, m � 0.8, K � 1, A � 0.5, h � 0.7143, and s � 0.672, and let the fear parameter C vary. Figure 11 illustrates that both predator and prey densities are in decline with the increase of fear parameter C.
T im e t S p a c e ϖ 16 Complexity

Conclusion
Population ecosystems presenting a tipping point are widely circulated, and it is crucial to dive into the tipping mechanism and develop tools for predicting the occurrence of tipping points.To achieve these goals, this article puts forward a predator-prey competitive model with fear delays and spatial difusion and focuses on the analysis of bifurcation-induced tipping events.By means of bifurcation theory, the tipping points of the evolution of population size can be predicted.Mechanisms, processes, and measures of tipping are explored.For the case without time delays, the Turing instability-induced-tipping is studied, and the specifc condition under which Turing instability occurs is derived.For the case of time delays, the Hopf bifurcationinduced tipping is analyzed, and the expression of tipping points is given clearly.Tis makes it possible to forecast the appearance of tipping points.In order to understand the internal mechanism of tipping more deeply, we can determine the direction of Hopf bifurcations by using the center manifold theorem for PFDEs.It is also found that many tipping points may exist in ecological competition systems, and the tipping occurs many times as the fear delay increases.Our future work will be devoted to the factor of defense and cross-difusion in predator-prey systems.