Spatiotemporal Evolution Characteristics of Time-Delay Ecological Competition Systems with Food-Limited andDiffusion

In this paper, we put forward a time-delay ecological competition system with food restriction and diffusion terms under Neumann boundary conditions. For the case without delay, the conditions for local asymptotic stability and Turing instability are constructed. For the case with delay, the existence of Hopf bifurcation is demonstrated by analyzing the root distribution of the corresponding characteristic equations. Furthermore, by using the normal form theory and the center manifold reduction of partial functional differential equations, explicit formulas are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Finally, some simulation examples are provided to substantiate our analysis.


Introduction
Based on the diversity of biological populations and ecosystems, various ecological competition systems have been established and widely studied (see [1][2][3][4][5]). In ecological competition systems, the interaction between populations is usually reflected by functional response functions. Results have shown that the predation relationship between populations greatly affects the dynamic behavior of predatorprey systems (see [6][7][8]). In fact, with the development of the economy, humans will harvest biological populations and develop related biological resources to obtain economic benefits, such as fisheries, forestry, and wildlife management (see [9][10][11]). In recent years, many scholars have introduced the harvesting terms into biological systems to study the system modeling and related dynamic characteristics [12][13][14][15]. May et al. [16] established the following model to analyse the interaction between populations under different harvesting strategies: where U and V represent the prey and predator densities, respectively. r 1 and r 2 indicate the intrinsic growth rates. K represents the internal growth limit of the population without predators. α stands for the maximum predation coefficient of predator. β is the quality standard for measuring prey as food. H 1 and H 2 represent the impact of human harvesting for predator and prey populations. On this basis, scholars have studied the relevant dynamic characteristics of different harvesting terms in system (1). In [17], the authors studied the case of H 1 � r 1 h 1 u and H 2 � r 2 h 2 v in system (1), that is, the constant effort harvesting to both prey and predator population. Particularly, they analyzed the maximum sustainable yield of another population under the condition of limiting the harvest of one population. In [18], the authors discussed the case of H 1 � h 1 and H 2 � r 2 h 2 v for system (1), namely constant yield harvesting for prey and constant effort harvesting for the predator population. At the same time, the relevant dynamic characteristics of the system were analyzed and the effects of different management strategies on the stability of the system were compared. Moreover, in [19,20], the authors considered the constant yield harvesting of the prey population in system (1), that is, H 1 � h 1 and H 2 � 0. ey obtained the conditions of Bogdanov-Takens bifurcation and supercritical/subcritical Hopf bifurcation of codimension 1.
Based on the research of the appeal literature, it is found that only the impact of linear capture is taken into account. However, from the perspective of biology and economics, the linear harvesting term cannot accurately reflect the real social activities (harvesting population and developing biological resources) of humans and their impact on the predator-prey system. us, the nonlinear harvesting terms have been introduced to model the dynamics of predatorprey systems in recent years (see [21][22][23]). In [24], a realistic harvesting functional form was proposed as follows: where q represents the catchability coefficient and E stands for the harvesting effort. e conditions for Hopf bifurcation were derived and a nonlinear state feedback controller was designed to control the Hopf bifurcation. Consider the logistic equation Here, r is the intrinsic growth rate and C represents the carrying capacity. Meanwhile, we can see from (3) that the predicted relation of the specific growth rate, dN/N dt, to mass, N, is a straight line. However, when considering a measure of the portion of available limiting factors not yet utilized by the population, the linear average growth rate cannot accurately describe the growth trend of the population. Research shows the growth and development of organisms depend on the availability and utilization of food in the living environment, which implies that different degrees of food supply rates will affect the stable age composition of the population and then have an influence on the average growth rate of the population. us, when growth limitations are based on the proportion of available resources not utilized, Smith [25] established a food-limited growth function where c is positive constant and represents the rate of replacement of mass in the population at saturation. It follows from (4) that the predicted relation of the specific growth rate, dM/Mdt, to mass, M, is not a straight line but a concave curve.
Considering the growth limit is based on the proportion of unused available resources and setting H 1 � qEu/1 + fE, H 2 � 0, then system (1) becomes where F(u, v) represents the functional response function between populations. p represents that when the density of prey population is low, the predator population can switch to other prey for capture. en, we will analyse the cases on F(u, v) � βuv and F(u, v) � e 1 uv/e 2 + u in the discussion that follows, respectively. In nature, the survival and development of a population often depend on the amount of food and the living space available. Importantly, the greater the population density, the higher the requirements for the living environment. At the same time, the acquisition of food mainly depends on the living environment of the population, which shows that the change of the population's living space affects the survival and development of the population to a great extent. erefore, the population will instinctively migrate and diffuse in space to seek a more suitable environment for survival and development. It is necessary to consider the influence of the diffusion effect on population dynamics in predator-prey systems. Mathematically, the nonlinear system with diffusion will show complex dynamic properties [26][27][28][29]. In the reaction-diffusion system proposed by Turing [30,31], the spatial heterogeneity caused by the internal reaction-diffusion characteristics of the system results in the loss of system symmetry and makes the system self-organize to produce some spatial patterns. e process of pattern formation is called Turing instability (Turing bifurcation). e symmetry of the system is broken, leading to the formation of Turing patterns. erefore, we call this phenomenon "Turing instability caused by diffusive reaction" [32].
On the other hand, time delay has become a factor that cannot be ignored in many biological dynamic systems. A large number of studies have revealed that time delay has an important impact on the dynamic characteristics of biological systems and it is common in predator-prey systems, mainly including mature time delay, capture time delay, and pregnancy time delay [33][34][35]. 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 to the equilibrium state. In particular, Hopf bifurcation is a dynamic bifurcation phenomenon, which shows that when the parameters change near the critical value, the stability of the equilibrium point will change and periodic solutions will be generated in its small neighborhood. Meanwhile, it is found that time delay as a bifurcation parameter can induce Hopf bifurcation [29,36,37]. erefore, in this paper, we consider the pregnancy delay of the predator population and analyse the dynamic characteristics of ecological competition systems.

Complexity
Based on the discussion above, we introduce the diffusion effect of the population [26][27][28][29] and the pregnancy delay of the predator population [33][34][35] into system (5) to explore its impacts on the dynamic characteristics of the ecological competition system, which can be described by with Neumann boundary conditions and initial conditions Here, u(t, x), v(t, x) stand for the population density of prey and predator at a spatial location x and time t, respectively. D 1 , D 2 > 0 represent the diffusion coefficients associated to u and v, respectively. Δ denotes the Laplacian operator in R n . Suppose Ω � (0, π) is a bounded domain with a smooth boundary zΩ.

Lemma 1.
For any solution of system (6) without delay, Proof. Let (u(t, x), v(t, x)) be a nonnegative solution of system (6) without delay. Note that the functional response F(u, v) > 0. en, We can estimate the upper limits of u(t, x) and v(t, x) due to the standard comparison principle: In other words, for arbitrary ε 1 > 0, ε 2 > 0, there exists positive constants t 1 , t 2 such that for t ≥ t 2 , x ∈ Ω. So, the conclusion follows immediately. Compared with the models proposed in [1][2][3][4][5], model (6) is a new measure of the population with a nonlinear average growth rate based on food restriction. At the same time, we consider the reaction-diffusion factors for system (6) to study their influence on the dynamic behaviors of systems. In addition, on the basis of references [16][17][18][19][20], we introduce the nonlinear harvesting term of the harvesting-effort coefficient E into system (6) and study the stability and related dynamic characteristics of the system under Holling I and Holling II functional response functions. Importantly, the addition of pregnancy delay can more accurately reflect the evolution of the population and make the system show more complex dynamic characteristics than the model without delay, which is also a widely concerned direction in the research of biological systems. e main contributions of this paper can be stated as follows. In Section 2, the effects of diffusion on the dynamic behavior of the systems without time delay are investigated and some conditions for system stability and Turing instability are determined. It is found that the appropriate diffusion coefficients will lead to Turing instability. In Section 3, we analyse the stability of equilibrium and Hopf bifurcation in the predator-prey system with time delay as the bifurcation parameter. e condition for Hopf bifurcation is constructed and the expression of the bifurcation threshold is given. In Section 4, we calculated the direction of the bifurcations to get more information about the bifurcations. Section 5 uses some numerical examples to verify the correctness of the previous derivation. Section 6 gives the conclusion of this paper.

Equilibrium Stability and Turing Instability Analysis
Assume that the predator-prey relationship in system (6) satisfies Holling type I functional response, that is F(u, v) � βuv, then we make the following nondimensional scaling transformation [22]: Dropping the tildes, system (6) can be rewritten by .
Consider the case of no time delay in system (9), namely, τ � 0, then system (9) becomes Considering the practical significance of the ecosystem, we are interested in the coexisting equilibrium. In order to obtain the positive equilibrium of system (10), let en system (10) has a positive equilibrium E * � (u * , v * ), where v * � θ(p + u * )/b and u * satisfies the following quadratic equation: where A 0 � αn > 0, Lemma 2. For equation (12), we come to the following results: (1) If A 2 > 0, then equation (12) has no positive roots.
It follows from [38] that the Laplacian operator − Δ has the eigenvalue k 2 (k ∈ 0, 1, 2, . . . { }) under the homogeneous Neumann boundary condition. And the corresponding k�0 construct a basis of the phase space X and X is defined by with the inner product 〈·, ·〉. us, for system (10), the characteristic equation at E * is where Hence, Proof. It follows from Lemma 3 that a 11 < 0 when A 2 < 0. Notice that a 12 < 0, a 21 > 0 and a 22 < 0. For equation (14), we obtain Complexity 5 us, all roots of (25) have negative real parts for k ∈ N 0 , which implies that E * of system (10) is asymptotically stable.
□ Remark 1. Assume that there are no time delays and F(u, v) � βuv for system (6). It can be seen from eorems 1 and 2 that when A 2 < 0, the introduction of diffusion terms does not change the stability of E * , which means that Turing instability does not occur.
Next, we consider the predator-prey relationship among populations in system (6) satisfying Holling type II functional response, that is F(u, v) � e 1 uv/e 2 + u, where e 1 represents the maximum per capita reduction rate of the prey population and e 2 stands for the average saturation rate [3]. en we make the following nondimensional scaling transformation [22]: Dropping the tildes, then system (6) without time delay turns into System (15) has a positive equilibrium E * � (u * , v * ), where v * � θ(p + u * )/b and u * satisfies the following quadratic equation: where Lemma 4. For equation (16), we have the following results: (16) has a unique positive root Theorem 3. For system (15), we come to the following results: where us, the characteristic equation is where Obviously, for (36), we have We make the following assumptions: Theorem 4. If (H1) − (H3) hold and Y 2 < 0, then system (15) is locally asymptotically stable at E * .

Hopf Bifurcation Analysis
In this section, we consider the effect of time delay on the dynamics of the system and get the conditions for Hopf bifurcation of system (9).
Linearizing system (9) us, the characteristic equation is where I stands for 2 × 2 identity matrix and where Remark 3. It should be pointed out that this paper adopts the linearization method [29,36,37,39,40] to deal with the dynamics analysis of system (9), including the local stability, Turing instability, and Hopf bifurcation. It is common knowledge that Lyapunov's second method is important to the 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 diffusion proposed in this paper. e Lyapunov stability criterion can only give a sufficient condition for the stability of a system. In this paper, not only the condition of the local stability is established, but also the boundary of stability (the onset of Hopf bifurcation) is determined.

Direction and Stability of Hopf Bifurcation
e previous analysis has shown that system (9) admits a series of periodic solutions bifurcating from the trivial uniform steady state E * at some critical values. en, in this section, we are concerned with the direction of Hopf bifurcations and the stability of bifurcating periodic solution.

Numerical Simulations
In the following, we will carry out numerical simulations of three impacts of diffusion, time delay, and harvesting effort to illustrate our theoretical findings in the previous sections.
en we set D 1 � 0.1, D 1 � 0.8 and other variables are the same as above.

e Impact of Harvesting Effort.
Considering the impact of harvesting efforts, we fix α � 0.6, b � 0.4, h � 0.134, f � 0.8, p � 0.3, θ � 0.6, D 1 � 0.1, D 2 � 0.1 and let E vary in [0. 5,5]. e stability and instability regions for system (9) are depicted by mapping the nonlinear harvesting E to the critical value τ 0 0 in Figure 13. Obviously, the critical value τ 0 0 of bifurcation increases with the increase of E, that is, the stable region of system (9) is expanded with the increase of E.

Conclusion
We investigate the spatiotemporal dynamic evolution of a time-delay ecological competition system with food restriction and diffusion terms under Neumann boundary conditions. e conditions of asymptotic stability and Turing instability at the positive equilibrium of delay-free systems are obtained under different functional response functions. Compared with the classical population growth model described by the logistic equation, the model with food restriction studied in this paper is more in line with actual biological competition systems. e results show that the diffusion phenomenon caused by the change of population position in space will seriously affect the stability of biological competition systems, eventually resulting in the appearance of Turing instability. en, by selecting the time delay as the bifurcation parameter, we reveal that the delay can cause very complex dynamic phenomena. When the delay is less than the bifurcation critical value, the system maintains asymptotic stability at the positive equilibrium point, while the system becomes unstable and produces a

Complexity
Hopf bifurcation when the delay is greater than the bifurcation critical value. Meanwhile, by using the central manifold method, we derive the conditions for determining the bifurcation direction and the stability of the bifurcation periodic solution. e harvesting effort has a major influence on the stability and Hopf bifurcation. As the harvesting effort increases in the appropriate range, the bifurcation critical value increases; that is, the stable region of the system is expanded. erefore, we can conclude that the appropriate harvesting of biological populations and the rational development of biological resources can not only meet human economic needs but also play a beneficial role in biological systems.
Data Availability e data in relation to the findings of this study are available upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.