Spatial Dynamics of a Leslie–Gower Type Predator-Prey Model with Interval Parameters

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Introduction
Te increasing interest in mathematical modeling of ecosystems has heightened the need for understanding the dynamic behaviors of species. Of particular interest are the spatial dynamics of the prey-predator model, one of which is the following mite model introduced by Wollkind et al. [1,2], whose work was based on the following [3]: Te predator's and prey's population densities are denoted by V(T) and U(T), where R, S, K, H are all positive parameters. R is referred to as the intrinsic growth rate of the prey without predation. S is the intrinsic growth rate of the predator. Te feature of this model is that the prey's growth rate is logistic growth with capacity K. Yet, the predator's growth rate is interpreted as "logistic growth" with "carrying capacity" HU, which is called the Leslie-Gower term [4,5]. HU estimates the decrease in predator population density caused by the lack (per capita U/V) of its favorite prey, where H refers to the conversion rate into the predator's biomass. Although the predator can convert to the substitute of its preferred resource, its population density will sufer due to the availability of its favorite food U. P(U), termed as the functional response with the meaning of prey consumption rate of per predator at prey density U and predator density V, can be chosen as Holling type I-III functions [6][7][8][9]. Particularly, Holling type III function is more suitable for vertebrate species, i.e., P(U) � MU 2 /AU 2 + 1, where M and A are referred to as half-saturation constant and capture rate. Te process that the predators are able to change their predation rate and convert from less common to more common prey is the learning process. Te learning process can make the foraging system more efcient because the predators can switch to alternative food resources when they know that their population fuctuates. Holling III functional response is for this situation [10].
On the other hand, difusion is one of the spatial motions in the prey-predator model, which depicts the random mobility of individual species. By incorporating spatial motion, model (1) becomes the spatial predator-prey model of the Leslie-Gower type. Although there are many papers about the spatial prey-predator model of the Leslie-Gower type, such as bifurcation analysis or global stability of steady states, and so on, we mainly discuss spatial patterns [11][12][13][14][15]. Tere have been plenty of studies about spatial patterns of this model [16][17][18][19][20][21][22]. For the spatial prey-predator model of the Leslie-Gower type, some researchers focused on the Holling-Tanner type, where P(U) is chosen as Holling type II response function. Tey mathematically analyzed the Turing instability of the model and numerically simulated diferent kinds of patterns [23,24]. Some papers discussed specifc efects on pattern formation of the model, such as cannibalism, delay efects, and Allee efect [25][26][27]. Other papers discussed the efects of the cross-difusion [28][29][30].
Yet, difusion rate in above papers mentioned is constant. In fact, dispersal or difusion behavior among species is not simple movement, but is a complex process which can be normally divided into three stages: emigration, movement between areas, and immigration. Moreover, dispersal rates in the process of three stages are condition-dependent and are density-dependent in particular [31]. Te densitydependent dispersal rate can be either positive or negative. Te common species with positive or negative density-dependent dispersal rate are insects, birds or mammals in the real word [32,33]. Ten, based on the work above, researchers have studied spatial dynamics of difusive system with varying difusion rate [34,35]. However, coefcients in above papers are accurate, and all the spatial predator-prey models mentioned above fail to view the biological parameters as imprecise parameters. In reality, the impreciseness can happen because of incomplete information during the determination of the initial value, measurement, and data collection. After Zadeh's pioneering work of introducing uncertainty into mathematically modeling [36], more and more researchers have incorporated impreciseness in the mathematical models via fuzzy sets [37][38][39]. As far as predator-prey models are concerned, although there are researches about the imprecise ODE model or imprecise difusive model [40][41][42][43][44], few papers are about patterns of spatial model with interval difusion rate. Motivated by the work [34,35,44], we will study Turing patterns of an imprecise prey-predator model of the Leslie-Gower type with Holling type III function in this paper. For simplicity, we set up the corresponding accurate model frst: where Ω, a bounded domain, satisfes Ω ⊂ R 2 , and n is the outward unit normal vector of the smooth bound of zΩ. Te coefcients of difusion of prey and predator are D 1 , D 2 . By introducing dimensionless variables x � U/K, y � MKV/R, and t � RT, system (2) can be simplifed as where η � S/R, e � AK 2 , c � R/HMK 2 , d 1 � D 1 /R, and d 2 � D 2 /R. In Section 2, interval number and interval-valued function are introduced, and the imprecise model corresponding to model (3) is set up. Turing instability and amplitude equations are analyzed mathematically in Section 3. Teoretical analysis is illustrated by numerical simulations in Section 4. At last, conclusions about the model are drawn in Section 5.

Prerequisites and Modeling
In this section, we introduce interval numbers, intervalvalued function, and arithmetic operations about it. Ten, we set up the imprecise predator-prey model of the Leslie-Gower type.

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Discrete Dynamics in Nature and Society

Imprecise Leslie-Gower Type Predator-Prey Model with
Difusion. Assume that parameters in system (3) are imprecise and presented as interval numbers. Let η, e, c, d 1 , and d 2 be the interval parameters corresponding to η, e, c, d 1 , and d 2 . We have the imprecise predator-prey model of the Leslie-Gower type with difusion.
which is a continuous and strictly increasing function. Based on Teorem 1 in [40], the imprecise form of model (4) is given by

Bifurcation Analysis
In this section, we analyze the existence of positive equilibrium of model (5) frst and then fnd the conditions that ensure the appearance of the Turing pattern. (5). Te corresponding nondifusion model of (5) is

Analysis of Positive Equilibrium of Model
Obviously, model (6) has an equilibrium E 0 � (1, 0). We are interested in the interior equilibrium points where Te numbers of positive equilibria of (6) are the same as the numbers of positive real roots of H in I 0 � (0, 1). According to [45], (7) has at most three roots and at least one root in the interval I 0 � (0, 1), so system (6) has at most three equilibria and at least one equilibrium. Suppose E * � (x * , y * ) is any positive equilibrium of model (6), at which the Jacobian matrix is calculated by is stable when the following conditions hold: And where tr(J) and det(J) are trace and determinant of matrix J given by (5). We obtain the conditions for Turing instability in this subsection. Suppose E * � (x * , y * ) is one of the positive and steady equilibrium, whose characteristic polynomial is

Analysis of Turing Instability of Model
where the expression of J k is where k is the wavenumber. Te eigenvalue of (13) can be obtained by the following equation: where where Ten, the eigenvalue can be expressed as Re(λ k ) > 0 guarantees Turing instability. Clearly tr k > 0, the necessary condition for Turing instability is Δ k < 0 for some value of k. Tat is, And As a result, if (9), (10), (18), and (19) hold, then Turing patterns of model (5) can appear. (5). In order to study the efects of interval parameters on pattern selection, we choose d 1 as controlled number to deduce amplitude equations of (5) via multiple-scale analysis.

Pattern Selection of Model
Around (5) can be rewritten as where where A x j and A y j are the amplitudes of the pattern of x and y, c.c. denotes the complex conjugate item of And 4 Discrete Dynamics in Nature and Society Te detailed expression of derivatives of all orders of f, g to x, y can be seen (A.1)-(A.10) in Appendix.
Near Turing bifurcation threshold, d T 1 , the bifurcation parameter ( � d 1 ) 1− q (d 1 ) q , the variable X, time t, and the nonlinear term N can be expanded as where Te linear operator L is dissembled via Taylor expansion as follows: where And Te derivation of the amplitude is Substitute (24)-(29) into (20) and collect ε, ε 2 , and ε 3 . Following linear systems can be given: Discrete Dynamics in Nature and Society 5 where And Solving (30), the solution is as follows: c denotes the complex conjugate item of W 1 e ik 1 ·z + W 2 e ik 2 ·z + W 3 e ik 3 ·z and W j is the amplitude of the pattern with the mode exp(ik j · z) under the frst order perturbation, the form of which is determined by higher order term. Substituting equation (35) to (31), one can obtain the explicit expression of F j x , and F j y (j � 1, 2, 3) in (A.11) Appendix. From the Fredholm solvability condition [13], we know that if the vector function of the right side of (31) is orthogonal to the eigenvectors of the zero eigenvalue of the adjoint operator of L T , which can be denoted by L + T , there exists a nontrivial solution of (31). Substituting (35) into right side of (31), the eigenvector of the operator L + T is given as follows: In the light of the Fredholm solvability condition, the orthogonality condition is Ten, (37) yields where W j is the conjugation of W j (j � 1, 2, 3). Let the form of the solution of (31) be Employing the Fredholm solvability condition in (32) again, we have where Y j is the conjugation of Y j (j � 1, 2, 3). Multiplying (38) and (40) by ε and ε 2 , in consideration of (29) and A j � A x j � lA y i (j � 1, 2, 3), the following amplitude equations are given: where A j is the conjugation of A j (j � 1, 2, 3), and where the expression that E, D 1 , and D 2 stand for can be seen (A.20)-(A.24) in Appendix. According to [13], we get the results of pattern selection in Table 1.

Numerical Simulations
In this part, we numerically simulated patterns of system (5) in 2-dimensional and 200 × 200 space area, employing Euler-forward fnite diference method and zero-fux boundary condition. We chose space step size as Δh � 1 and time step size as Δt � 0.05 with time interval [0, 8000] . For the sake of similar patterns of both prey and predator, the prey's pattern was chosen to investigate in detail, and the simulations will not stop until the patterns reach their steady state. Additionally, we varied the interval variable q to show how the interval parameter would afect the spatial dynamics of system (5).

Efects of Interval Parameters on Controlled Parameter and Pattern Selection.
In this section, we selected all parameters as interval parameters (Table 2) and studied the efects of interval parameters on controlled parameter and pattern selection.
It can be seen from Table 3 that the values of controlled parameter d T 1 increase from 0.2860078311 to 0.3264191705 as the values of interval variable q increase. According to the defnition of interval number, it means that when parameters of system (5) are interval numbers, the controlled parameter is also an interval number with the value [0.2860078311, 0.3264191705]. By the same way, the boundary μ 3 and μ 4 of the domain of pattern selection (Table 1), where the controlled parameter lies, are also interval numbers. Moreover, the domains, i.e., (μ 2 , μ 3 ) and (μ 3 , μ 4 ), of pattern selection expand as an interval variable q increases. Figures 1-3 show the patterns that appear in corresponding pattern selection domain when q � 0.3, q � 0.6 and q � 0.9. All the patterns in Figures 1-3 are in agreement with the theoretical Discrete Dynamics in Nature and Society analysis in Table 1, i.e., spot patterns (Figures 1(a), 2(a), and 3(a)) appear when μ lies in (μ 2 , μ 3 ), and mixture of spots and stripes (Figures 1(b), 2(b), and 3(b)) emerge when μ lies in (μ 2 , μ 3 ), and stripe patterns ( (Figures 1(c), 2(c), and 2(c)) arise when μ > μ 4 . From this, we conclude that interval parameter of system (5) makes the boundary μ 3 and μ 4 be an interval number or changes the range of pattern selection domain (μ 2 , μ 3 ) and (μ 3 , μ 4 ), but it does not change the types of patterns in the corresponding pattern selection domain.

Efects of Interval Parameter d 1 on Patterns of System (5).
Te difusion coefcient is an important parameter with the meaning of random motion, so we fgured out what Types of the pattern Stability μ 2 < μ < μ 3 Spot pattern Unstable μ 3 < μ < μ 4 Mixture of spots and stripes pattern Stable μ > μ 4 Stripe pattern Stable      Table 4, where we selected d 1 as interval parameters and others as accurate.
It can be seen from Figure 4 that the microstructure of spot pattern of the prey changes with the increasing of the interval variable q: the density of the cold spot pattern gradually becomes sparse. Specifcally, there are fewer and fewer isolated areas formed by the low density of the prey. To diferentiate the patterns in Figure 4 more clearly, we plotted time-series of prey's density at spatial location (21,21) in Figure 5 and spatial mean density of the prey with respect to interval variable q in Figure 6. From Figure 5, when t > 200, the density of prey lies in (0.4, 0.45) with q � 0 in Figure 5 Figure 5(e). One can see that the density of the prey decreases along with the increasing of q, which is in accordance with results of Figure 6. From above analysis, it means that the interval parameter d 1 not only afects the microstructure of pattern of the prey but also the density of the prey.

Efects of Interval Parameter c on Patterns of System (5).
Te parameter c denotes the reliance of predators on prey, which is an intrinsic factor of predator-prey systems of the Leslie-Gower type, so we studied the efects of the interval   Discrete Dynamics in Nature and Society parameter c on model (5). For this purpose, we chose c as the interval parameter and others as accurate, the values of which were provided in Table 5. Ten, we simulated patterns of prey numerically. From Figure 7, patterns of the prey transit orderly with the increasing of the interval variable q: hot spot pattern (Figure 7(a)), mixture of hot spot and stripe pattern ( Figure 7(b)), stripe pattern (Figure 7(c)), mixture of cold spot and stripe pattern (Figure 7(d)), cold spot pattern (Figure 7(e)) with the increasing of the interval variable of q. It is easy to see the diferences of patterns in Figure 7. Compared with Figure 4, Figures 8 and 9 show that the changing of prey's density show an opposite trend, i.e., the density of the prey increases as the values of the interval    variable q grows. We conclude that the interval parameter c has a remarkable efect on types of pattern and the density of the prey.

Discussion and Conclusions
Te spatial dynamics of a prey-predator model with an interval parameter were initially studied in this paper, compared with the former work [23][24][25][26][27][28][29][30]. First, conditions of Turing instability of the model with an interval parameter were gained via an interval-valued function and mathematical analysis. Ten, the efects of all or some interval parameters on the spatial dynamics of system (5) were discussed. Specifcally, via multiple-scale analysis and numerical simulation, pattern selection and pattern transitions for system (5) under interval parameters were obtained. It is well known that multiple-scale analysis is efective when the values of parameters are very close to the controlled parameter. According to our results, whatever the sensitive parameter, the controlled parameter is also an interval number. Te results in Section 4.2 and 4.3 show that the efects of diferent interval parameters on system (5) are diferent. In this paper, the impreciseness is presented by the interval parameter, so we conclude that the impreciseness of the parameter does afect the behavior of the entire spatial biological system. Based on the conclusion, our hypothesis proposed in the introduction that researchers should incorporate impreciseness when they make mathematical modeling is reasonable.
It should be noted that we choose d 1 as a controlled number to study the efects of interval parameters on pattern selection. Other parameters can also be chosen as controlled parameters. Moreover, the interval variable q describes any position of the value of an interval-valued function on the closed interval that presents the interval number. According to the defnition of an interval-valued function, the bigger value of q means the value of the interval-valued function lies closer to the upper limit of the interval number. Our results reveal the tendency of the spatial dynamics in the process of the values of an interval-valued function increasing to the upper limit of the interval number (see Tables 6 and 7). Tis study has concentrated on the efects of interval number on the spatial dynamics of a spatial prey-predator model. Despite its preliminary nature, imprecise parameters make the model closer to the reality of a lack of accurate information and provide a new perspective on studying spatial predator-prey models. In future work, it intrigues us to think about the efects of impreciseness on the spatial epidemic model.