# Complex Dynamics of a Ratio-Dependent Predator-Prey Model Induced by Spatial Motion

Authors: Caiyun Wang; Jing Li; Ruiqiang He
Journal: Complexity
 (2021)

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## Abstract

One of the most efficient predator-prey models with spatial effects is the one with ratio-dependent functional response. However, there is a need to further explore the effects of spatial motion on the dynamic behavior of population. In this work, we study a ratio-dependent predator-prey model with diffusion terms. The aim of this work is to investigate the changes in predator’s distribution in space as the prey populations change their mobility. We observe that the frequency diffusion of the prey gives rise to the sparse density of the predator. Moreover, we also observe that the increasing rate of the conversion into predator biomass induces pattern transitions of the predator. Specifically speaking, Turing pattern of the predator populations goes gradually from a spotted pattern to a black-eye pattern, with the intermediate state being the mixture of spot and stripe pattern. The simulation results and analysis of this work illustrate that the diffusion rate and the real intrinsic factor influence the persistence of the predator-prey system mutually.


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## Body

## 1. Introduction

The dynamical behavior of the prey-predator model has been investigated by researchers for many years after the pioneering work was accomplished by Lotka and Volterra. The authors present the well-known Lotka–Volterra predator-prey model. It is established that the response function is the key element in the predator-prey models. The ratio-dependent functional response was first proposed by Leslie and Gower in 1973 [1, 2]. The authors discuss that the predator’s growth is described by prey’s density, as well as by the ratio of density of the predator to prey. Afterwards, the Holling–Tanner rate in the predator-prey model is presented in the literature. This model is also known as the Holling–Tanner predator-prey model [3]. The dynamics such as local and global stability, limit cycles, and pattern solutions of Holling–Tanner predator-prey model have already been investigated [4–7]. The factors such as delay and random are also considered in the recent literature [8–13]. The fuzzy approach is also considered to construct the predator-prey model [14, 15]. However, the Holling–Tanner-type predator-prey models are important because of their appropriateness for modeling the real ecological interactions between predator and prey species. The Holling–Tanner predator-prey model is(1)dNdT=RN1−NK−MNPN+D,dPdT=PΘ1−HPN,where NT and PT denotes the prey and predator density, respectively, at time T.R,K,Θ,M,D, and H are all positive, and their definitions are given in Table 1.Table 1
The definition of parameters used in model (1).
ParameterMeaningRThe prey’s intrinsic growth rateKThe carrying capacity of the preyMThe largest amount of prey that predator can capture per unit timeΘPredator intrinsic growth rateHConversation rate into the biomass of predator by preyDA saturation parameterThe termMNP/N+D in model (1) is known as Holling type II response function [16] which was proposed by Holling in 1965. In addition, it is assumed that the functional response is expressed by Holling type III response function [17–19], which is more suitable for the population of vertebrates. The ratio-dependent predator-prey model with Holling type III functional response is(2)dNdT=RN1−NK−MN2PAN2+1,dPdT=PΘ1−HPN,where A is the half-saturation constant.On the other hand, there are a large number of organisms with spatial pattern distributions [20–22]. The distribution of steady spatial pattern is called Turing pattern. This model was put forward when Turing presented the theory of mathematical explanation for the pattern formation of the animal skin [23]. Following Turing’s famous work, the researchers have already explained self-organized pattern formation in various fields such as geology, physics, chemistry, and biology [20–28]. Especially, for the diffusive predator-prey model in ecology, pattern formation helps us in a better way to understand the influence of predator’s individual mobilities, the dynamics of stability, and oscillations of the prey species. There are various works that present the spatial predator-prey models [7, 29–35]. Wu and Song discussed the joint impact of nonlocal term and delay on the spatial predator-prey model [36]. Cao and Jiang indicate that the nonlocal reaction term can stabilize the spatially inhomogeneous periodic solutions [37]. Incorporating the diffusive terms into model (2), we have(3)∂N∂T=RN1−NK−MN2PAN2+1+D1∇2N,x,T∈Ω×0,∞,∂P∂T=ΘP1−HPN+D2∇2N,x,T∈Ω×0,∞,∂Nx,T∂n=∂Px,T∂n=0,x,T∈∂Ω×0,∞,Nx,0=N0>0,x∈Ω,Px,0=P0>0,x∈Ω,where D11 and D22 represent the diffusion coefficients of the prey and predator, respectively, which are all positive.The mathematical analysis of the spatial predator-prey model used to be based on linear stability analysis. However, linear stability analysis is only meaningful in the small domain of the equilibrium, where the linear system is approximated to nonlinear systems. The multiple scale method [38–40] is employed to study spatial dynamic behaviors of the model in our present paper. This method is effective when the system under consideration is a partial differential equation. Specifically, the parameters in system are decomposed into different scales. The amplitude equation is an important tool to understand the pattern selection near the Turing bifurcation boundary. By solving the amplitude equation, we obtain the domain of the controlled parameter, where the steady state pattern arises. The rest of our paper is organized as follows.In Section2, we present an example to show the conditions for the occurrence of Turing instability by bifurcation analysis. In Section 3, we derive the amplitude equations of our model on the basis of multiple scale analysis. In Section 4, we present the simulations performed to illustrate the different patterns in the corresponding Turing domain. Moreover, the predator’s pattern transitions driven by prey’s diffusion rate and conversation rate of biomass into predator are discussed in detail. Finally, in Section 5, we conclude our work.


## 2. Bifurcation Analysis

In this section, we present the conditions for the occurrence of Turing pattern. We choose the scaleu˜=N/K,v˜=MKP/R,andt˜=RT, and we obtain(4)∂u˜∂t˜=u˜1−u˜−u˜2v˜eu˜2+v˜+d1∇2u˜,x,t˜∈Ω×0,∞,∂v˜∂t˜=v˜η1−γv˜u˜+d2∇2v˜,x,t˜∈Ω×0,∞,∂u˜x,t˜∂n=∂v˜x,t˜∂n=0,x,t∈∂Ω×0,∞,u˜x,0=u0>0,x∈Ω,v˜x,0=v0>0,x∈Ω,where η=Θ/R,e=AK2,γ=HR/MK2,d1=KD1/R,andd2=RD2/M2K3.The analysis of the existence and local stability of the positive equilibrium is presented in [41]. Here, we present an example to illustrate the existence and local stability of the positive equilibrium. Let E∗u˜∗,v˜∗ be an interior steady equilibrium of model (4). The Jacobin matrix of model (4) is(5)J=fu˜fv˜gu˜gv˜,u˜1−u˜−u˜2v˜eu˜2+1≜fu˜,v˜,v˜η1−γv˜u˜≜gu˜,v˜,and the trace and determinant of matrix J are(6)trJ=−1+eu˜∗2−2eu˜∗31+eu˜∗2−η,detJ=ηu˜∗1+eu˜∗2F′u˜∗.On the basis of parameters presented in Table2, we calculate the equilibrium and get the stability of the E∗u˜∗,v˜∗.Table 2
An example to illustrate the existence and stability of the equilibrium of model (4).
ηγeu˜∗v˜∗trJdetJ0.70.010402810361400.35133.74086284−0.37313053710.2532783223The characteristic polynomial atE∗u˜∗,v˜∗ is(7)λE−Jk=0,where(8)Jk=fu˜−d1k2fv˜gu˜gv˜−d2k2and k is the wave number. The eigenvalue is(9)λk2−trkλk+Δk=0,where(10)trk=fu˜+gv˜−k2d1+d2,Δk=fu˜gv˜−fv˜gu˜−k2d2fu˜+d1gv˜+d1d2k4=detJ−k2d2fu˜+d1gv˜+d1d2k4.Therefore, the eigenvalue of (7) is(11)λk=trk±trk2−4Δk2.The Hopf bifurcation occurs when the conditionsReλk=0 and Imλk≠0 hold at k=0. However, Turing bifurcation occurs when conditions Reλk=0and Imλk=0 hold at k=kT>0. And, kT satisfies(12)kT2=fu˜gv˜−fv˜gu˜d1d2=detJd1d2.Substituting (12) into Δk=0, we get the expression of d1T. The value of this expression represents the critical value of Turing bifurcation parameter. Turing domain and the real part of eigenvalues of model 4 are shown in (Figures 1 and 2).(13)d1T=d2fu˜2fu˜gv˜−fv˜gu˜+fv˜2gu˜2−fu˜fv˜gu˜gv˜/fu˜−gv˜.Figure 1
Turing domain of model (4).Figure 2
Dispersion relation of system (4), which shows the real part of eigenvalues of model (4).


## 3. Multiple Scale Analysis

We choosed1 and d1T as a control parameter and the bifurcation threshold, respectively. This is done in order to deduce the amplitude equations via multiple scale analysis [42, 43] and judge the type of the Turing pattern.Neard1=d1T, system (4) is written at E∗u˜∗,v˜∗ as follows:(14)∂U∂t=LU+N,where(15)U=u˜v˜,L=fu˜+d1∇2fv˜gu˜gv˜+d2∇2,(16)h˜3=16p111u˜13+16p222v˜13+16p112+p121+p211u˜12v˜1+16p122+p212+p221u˜1v˜12+p11u˜2u˜1+p22v˜2v˜1+12p12+p21u˜2v˜1+u˜1v˜216q111u˜13+16q222v˜13+16q112+q121+q211u˜12v˜1+16q122+q212+q221u˜1v˜12+q11u˜2u˜1+q22v˜2v˜1+12q12+q21u˜2v˜1+u˜1v˜2,where pijkandqijk denote the derivative of ith, jth, and kth variable successively to fandg at E∗u˜∗,v˜∗.U,t, and N are expanded with a small parameter ε.(17)d1T−d1=εd11+ε2d22+…,t˜=T0+εT1+ε2T2,U=u˜v˜=εu˜1v˜1+ε2u˜2v˜2+ε3u˜3v˜3+oε4,N=ε2h˜2+ε3h˜3+oε4,where(18)h˜2=12p11u˜12+12p12+p21u˜1v˜1+12p22v˜1212q11u˜12+12q12+q21u˜1v˜1+12q22v˜12,h˜3=16p111u˜13+16p222v˜13+16p112+p121+p211u˜12v˜1+16p122+p212+p221u˜1v˜12+p11u˜2u˜1+p22v˜2v˜1+12p12+p21u˜2v˜1+u˜1v˜216q111u˜13+16q222v˜13+16q112+q121+q211u˜12v˜1+16q122+q212+q221u˜1v˜12+q11u˜2u˜1+q22v˜2v˜1+12q12+q21u˜2v˜1+u˜1v˜2.Meanwhile, the linear operatorL is dissembled by Taylor expansion and (14) transforms to(19)L=LT+d1T−d1M,where(20)LT=fu˜+d1T∇2fv˜gu˜gv˜+d2∇2,M=−∇2000.The derivation of the amplitude is(21)∂A∂t˜=ε∂A∂T1+ε2∂A∂T2+….We substitute (17)–(21) into (14) and collect the ε,ε2, and ε3, represented by the following set of linear equations:(22)ε:LTu˜1v˜1=0,(23)ε2:LTu˜2v˜2=∂∂T1u˜1v˜1−d11Mu˜1v˜1−h˜2≜Fu˜Fv˜,(24)ε3:LTu˜3v˜3=∂∂T1u˜2v˜2+∂∂T2u˜1v˜1−d11Mu˜2v˜2−d22Mu˜1v˜1−h˜3≜Hu˜Hv˜,where(25)Fu˜Fv˜=∑j=13Fu˜jFv˜jeikj⋅r+c.c.and(26)HuHv=∑j=13HujHvjeikj⋅r+c.c..According to [43], solving (22), the solution is expressed as(27)u˜1v˜1=l1W1eik1⋅r+W2eik2⋅r+W3eik3⋅r+c.c.,where l=fu˜d2−gv˜d1T/2gu˜d1Tandkj=kT. c.c. represents the complex conjugate and the pattern of the mode expikj⋅r is Wjj=1,2,3 under the first-order perturbation.Substituting (27) into right-hand side of (23), the eigenvector of the operator LT+ is as follows:(28)1l2e−ikj⋅r+c.c.,j=1,2,3,where LT+ represents the eigenvector of the zero eigenvalue of the adjoint operator of LT and(29)l2=2fv˜d1Tfu˜d2−gv˜d1T.On the basis of Fredholm solvability condition, the orthogonality condition is given as(30)1,l2Fu˜jFv˜j=0,j=1,2,3.(30) yields(31)l+l2∂W1∂T1=d11kT2lW1+p11l2+p12+p21l+p22+l2q11l2+q12+q12l+q22W¯2W¯3,l+l2∂W2∂T1=d11kT2lW2+p11l2+p12+p21l+p22+l2q11l2+q12+q12l+q22W¯1W¯3,l+l2∂W3∂T1=d11kT2lW3+p11l2+p12+p21l+p22+l2q11l2+q12+q12l+q22W¯1W¯2.Let the solution of (23) have the following form:(32)u˜2v˜2=U˜0V˜0+∑j=13U˜jV˜jeikj⋅r+∑j=13U˜jjV˜jjei2kjr+U˜12V˜12eik1−k2r+U˜23V˜23eik2−k3⋅r+U˜31V˜31eik3−k1⋅r+c.c..Substituting (32) into (23) and merging the coefficients of exp0,expikj⋅r, expi2kj⋅r, and expikj−ki⋅r, we obtain the following resultant relation:(33)U˜0V˜0=u˜0v˜0W12+W22+W32,U˜j=lV˜j,U˜ijV˜ij=u˜11v˜11Wj2,U˜mnV˜mn=u˜mnv˜mnWmW¯n,where(34)u˜0v˜0=1fu˜gv˜−fv˜gu˜−gv˜p11l2+p12+p21l+p22+pv˜q11l2+q12+q12l+q22−fu˜q11l2+q12+q12l+q22+qu˜p11l2+p12+p21l+p22,u˜11v˜11=121fu˜−4KT2d1Tgv˜−4KT2d2−fv˜gu˜−gv˜−4KT2d2p11l2+p12+p21l+p22+pv˜q11l2+q12+q12l+q22−fu˜−4KT2d1Tq11l2+q12+q12l+q22+gu˜f11l2+p12+p21l+p22,u˜mnv˜mn=1fu˜−3KT2d1Tgv˜−3KT2d22−fv˜gu˜−gv˜−3KT2d2p11l2+p12+p21l+p22+fv˜q11l2+q12+q12l+q22−fu˜−3KT2add1Tq11l2+q12+q12l+q22+qu˜p11l2+p12+p21l+p22.We make use of the Fredholm solvability condition in (24). Now, we have(35)1,l2Hu˜jHv˜j=0j=1,2,3.From (35), we get(36)l+l2∂V1∂T1+∂W1∂T2=lkT2d11V1+d22W1+DV¯2W¯3+V¯3W¯2+E1W12+E2W22+W32W1,l+l2∂V2∂T1+∂W2∂T2=lkT2d11V2+d22W2+DV¯1W¯3+V¯3W¯1+E1W22+E2W12+W32W2,l+l2∂V3∂T1+∂W3∂T2=lkT2d11V3+d22W3+DV¯1W¯2+V¯2W¯1+E1W32+E2W1|2+W22W3,where(37)D=lr1+s1+l2lr2+n2,E1=br1+a1s1+P12+l2br2+b1s2+P22,E2=cr1+c1s1+P1+l2cr2+c1s2+P2,and(38)b=u˜0+u˜11,c=u˜0+u˜mn,b1=v˜0+v˜11,c1=v˜0+v˜mn,r1=p11l+12p12+p21,s1=p22+12p12+p21l,r2=q11l+12q12+q21,s2=q22+12q12+q21l,P1=p111l3+p112+p121+p211l2+p122+p212+p221l+p222,P2=q111l3+q112+q121+q211l2+q122+q212+q221l+q222.Multiplying (31) and (36) by ε and ε2, together with (21) and(39)Di=Diu˜=lDiv˜,i=1,2,3,we have the following amplitude equations:(40)τ0∂A1∂t=μD1+hD¯2D¯3−g1D12+g2D22+D32D1,τ0∂A2∂t=μD2+hD¯1D¯3−g1D22+g2D12+D32D2,τ0∂A3∂t=μD3+hD¯1D¯2−g1D32+g2D12+D22D3,where(41)τ0=l+l2d1TlkT2,μ=d1T−d1d1T,h=Dd1Tl2kT2,g1=−E1d1Tl3kT2,g2=−E2d1Tl3kT2.On the basis of aforementioned analysis, we obtain the following results about the pattern selection of system (4) shown in (Table 3).Table 3
The pattern selection of model (4).
μPatternStabilityμ2<μ=0.1115957587<μ3Spot patternStableμ3<μ=0.5064420882<μ4Coexistence of the spots and stripe patternsBistableμ=0.8025768353>μ4Tripe patternsStable


## 4. Numerical Simulations of Patterns

### 4.1. Pattern Selection

In this section, we select the suitable value ofd1 and perform numerical simulations to verify the theoretical results, i.e., to see whether the patterns arise in the corresponding Turing domain. We make use of the zero-flux boundary condition with 200×200 lattice sites. The space step is Δh=1, and the time step is Δt=0.01. For the sake of the similar type of patterns of the predator and prey, we choose the predator to exhibit our investigations. We start with initial spatial state as the heterogeneity space. The simulations are performed until the patterns do not change their characteristic anymore. The value of parameters is presented in Tables 4 and 5.Table 4
The parameters set in system (4).
ηγed20.60.0020567794325254Table 5
The parameters calculated in system (4).
d1Tg1g2hμ1μ2μ3μ41.01305234576.6130004171.12173944.675460751−0.0130473887300.18750263050.7938085466Figure3 shows the evolutions of spatial pattern of the predator at final instants when d1=0.95,d1=0.5,d1=0.47, and d1=0.2, respectively. As presented in Figure 3, we obtain three different patterns in the corresponding Turing domains, which are hot spot patterns, i.e., Figure 3(a), mixture of spots and stripes, i.e., Figures 3(b) and 3(c), and stripe patterns, i.e., Figure 3(d). Thus, the simulation results shown are aligned with the theoretical analysis. However, it is noticeable that there is a little difference between the two mixtures in Figure 3(c) and in Figure 3(d), because they have different wavelengths.Figure 3
The spatial patterns of the predator with different values of 31: (a) 31 = 0.95; (b) 31 = 0.5; (c) 31 = 0.47; (d) 31 = 0.2.
(a)(b)(c)(d)


### 4.2. Effect of the Prey’s Diffusion on Patterns of Predator

In this section, we discuss the pattern formation driven by the diffusion rate of the prey by varying the value of the prey’s diffusion, while the values of the other parameters are kept the same as in Section4.1.First, we fixd1=1.025 which is larger than d1T and transcends the bifurcation domain. Please note that there are no patterns emerging in Figure 4(a). This is in line with our theoretical results. As d1 becomes smaller than d1T decreasing from 1.009 to 0.83 in Figures 4(b)–4(d), the spot pattern gets more and more dense. In order to analyze the variation in density, we compare Figure 4(b) with Figure 4(d) at the same iterations in detail. On the one hand, from Figures 5 and 6, we find that Figure 6 transforms faster than Figure 5. For example, Figure 6(b) starts to become the spot pattern while Figure 5(b) is still irregular in shape after 3400 iterations. On the contrary, we observe that the spots in Figure 5(d) are sparser than in Figure 6(d). Thus, the density of the predator will decrease as there is an increase in the mobility of the prey. According to the Reichenbach’s viewpoint [44], we can illustrate our results effectively, i.e., when the prey species move frequently, the prey will encounter the predator easily. This leads to the decrease of the prey and aggravate the competition of the predator. As a result, the predators confront the danger of extinction. Moreover, we conclude that the pattern formation is induced by the diffusion rate of the prey by comparing Figure 4(a) with Figures 4(b)–4(d).Figure 4
The spatial patterns of the predator with different value of 31: (a) 31 = 1.025; (b) 31 = 1.009; (c) 31 = 0.9; (d) 31 = 0.83.
(a)(b)(c)(d)Figure 5
The iterations of stripe patterns of the predator with 31 = 1.009. The number of iterations is (a) 0, (b) 3600, (c) 130000, and (d) 150000.
(a)(b)(c)(d)Figure 6
The iterations of stripe patterns of the predator with 31 = 0.83. The number of iterations is (a) 0, (b) 3600, (c) 130000, and (d) 150000.
(a)(b)(c)(d)


### 4.3. Pattern Transitions of the Predator Driven by Parameterγ

We fit the values of parametersη,e,d1,andd2 presented in Table 6, while the value of γ gradually increases. We investigate the changes in the spatial distribution of the predator as the conversion rate into predator biomass varies. Specifically, when γ is fixed γ=0.007041633659, hot spot pattern dominates the spatial region in Figure 7(a). In Figure 7(b), when we have γ=0.008509501103, then the mixture of hot spots and stripes appears. As γ continues to grow to 0.009332994621, there is just the stripe pattern that is presented in Figure 5(c). Similarly, when γ increases to γ=0.01032204568, the mixture of cold spots and stripes arises, which is shown in Figure 7(d). Finally, when γ becomes larger, we simulate the cold spot pattern presented in Figure 7(e).Figure 7 shows the orderly changes of the pattern transition, as well as the density of the predator increases from 85 to 90 and then decreases to 85 again with an increase in the value of γ, because γ denotes the number of the prey species which are captured by a single predator. The higher the value of parameter γ, the more food the predator species eat. However, please note that there is an effect of the environmental capacity, so the number of the predators cannot increase forever. So, the population of the predator decreases again when it exceeds the threshold value. The simulation results of this work present a clear biological meaning.Table 6
The parameters selected in system (4).
ηed1d20.63500.24Figure 7
The transitions patterns of the predator induced by the conversion rate of the prey into predator biomass and the other parameters are set in Table6. Iterations: (a) 0.007041633659; (b) 0.008509501103; (c) 0.009332994621; (d) 0.01032204568; (e) 0.01092704597.
(a)(b)(c)(d)(e)


## 4.1. Pattern Selection

In this section, we select the suitable value ofd1 and perform numerical simulations to verify the theoretical results, i.e., to see whether the patterns arise in the corresponding Turing domain. We make use of the zero-flux boundary condition with 200×200 lattice sites. The space step is Δh=1, and the time step is Δt=0.01. For the sake of the similar type of patterns of the predator and prey, we choose the predator to exhibit our investigations. We start with initial spatial state as the heterogeneity space. The simulations are performed until the patterns do not change their characteristic anymore. The value of parameters is presented in Tables 4 and 5.Table 4
The parameters set in system (4).
ηγed20.60.0020567794325254Table 5
The parameters calculated in system (4).
d1Tg1g2hμ1μ2μ3μ41.01305234576.6130004171.12173944.675460751−0.0130473887300.18750263050.7938085466Figure3 shows the evolutions of spatial pattern of the predator at final instants when d1=0.95,d1=0.5,d1=0.47, and d1=0.2, respectively. As presented in Figure 3, we obtain three different patterns in the corresponding Turing domains, which are hot spot patterns, i.e., Figure 3(a), mixture of spots and stripes, i.e., Figures 3(b) and 3(c), and stripe patterns, i.e., Figure 3(d). Thus, the simulation results shown are aligned with the theoretical analysis. However, it is noticeable that there is a little difference between the two mixtures in Figure 3(c) and in Figure 3(d), because they have different wavelengths.Figure 3
The spatial patterns of the predator with different values of 31: (a) 31 = 0.95; (b) 31 = 0.5; (c) 31 = 0.47; (d) 31 = 0.2.
(a)(b)(c)(d)


## 4.2. Effect of the Prey’s Diffusion on Patterns of Predator

In this section, we discuss the pattern formation driven by the diffusion rate of the prey by varying the value of the prey’s diffusion, while the values of the other parameters are kept the same as in Section4.1.First, we fixd1=1.025 which is larger than d1T and transcends the bifurcation domain. Please note that there are no patterns emerging in Figure 4(a). This is in line with our theoretical results. As d1 becomes smaller than d1T decreasing from 1.009 to 0.83 in Figures 4(b)–4(d), the spot pattern gets more and more dense. In order to analyze the variation in density, we compare Figure 4(b) with Figure 4(d) at the same iterations in detail. On the one hand, from Figures 5 and 6, we find that Figure 6 transforms faster than Figure 5. For example, Figure 6(b) starts to become the spot pattern while Figure 5(b) is still irregular in shape after 3400 iterations. On the contrary, we observe that the spots in Figure 5(d) are sparser than in Figure 6(d). Thus, the density of the predator will decrease as there is an increase in the mobility of the prey. According to the Reichenbach’s viewpoint [44], we can illustrate our results effectively, i.e., when the prey species move frequently, the prey will encounter the predator easily. This leads to the decrease of the prey and aggravate the competition of the predator. As a result, the predators confront the danger of extinction. Moreover, we conclude that the pattern formation is induced by the diffusion rate of the prey by comparing Figure 4(a) with Figures 4(b)–4(d).Figure 4
The spatial patterns of the predator with different value of 31: (a) 31 = 1.025; (b) 31 = 1.009; (c) 31 = 0.9; (d) 31 = 0.83.
(a)(b)(c)(d)Figure 5
The iterations of stripe patterns of the predator with 31 = 1.009. The number of iterations is (a) 0, (b) 3600, (c) 130000, and (d) 150000.
(a)(b)(c)(d)Figure 6
The iterations of stripe patterns of the predator with 31 = 0.83. The number of iterations is (a) 0, (b) 3600, (c) 130000, and (d) 150000.
(a)(b)(c)(d)


## 4.3. Pattern Transitions of the Predator Driven by Parameterγ

We fit the values of parametersη,e,d1,andd2 presented in Table 6, while the value of γ gradually increases. We investigate the changes in the spatial distribution of the predator as the conversion rate into predator biomass varies. Specifically, when γ is fixed γ=0.007041633659, hot spot pattern dominates the spatial region in Figure 7(a). In Figure 7(b), when we have γ=0.008509501103, then the mixture of hot spots and stripes appears. As γ continues to grow to 0.009332994621, there is just the stripe pattern that is presented in Figure 5(c). Similarly, when γ increases to γ=0.01032204568, the mixture of cold spots and stripes arises, which is shown in Figure 7(d). Finally, when γ becomes larger, we simulate the cold spot pattern presented in Figure 7(e).Figure 7 shows the orderly changes of the pattern transition, as well as the density of the predator increases from 85 to 90 and then decreases to 85 again with an increase in the value of γ, because γ denotes the number of the prey species which are captured by a single predator. The higher the value of parameter γ, the more food the predator species eat. However, please note that there is an effect of the environmental capacity, so the number of the predators cannot increase forever. So, the population of the predator decreases again when it exceeds the threshold value. The simulation results of this work present a clear biological meaning.Table 6
The parameters selected in system (4).
ηed1d20.63500.24Figure 7
The transitions patterns of the predator induced by the conversion rate of the prey into predator biomass and the other parameters are set in Table6. Iterations: (a) 0.007041633659; (b) 0.008509501103; (c) 0.009332994621; (d) 0.01032204568; (e) 0.01092704597.
(a)(b)(c)(d)(e)


## 5. Conclusions

In this work, we study a ratio-dependent prey-predator model. The pattern selection of our model has been investigated via multiple scale analysis near the Turing bifurcation point. First, we obtain the condition of the Turing instability of our model. Then, by choosingd1 as a bifurcation parameter, we obtained the amplitude equations and presented the Turing pattern selection of model (4). The simulation results presented in this work are aligned with the theoretical analysis, i.e., H0 spot patterns arise when μ lies in domainμ2,μ3 with d1=0.95; the stable mixture of spots and stripes appears when μ3<μ<μ4 with d1=0.5 and d1=0.47, while only stripe pattern is simulated when μ>μ4 and d1=0.2. On the contrary, we study the effect of the spatial motion and conversion rate of predator biomass on density of the predator. The decreasing value of the former agent d1 causes the sparse density in the predator population, while the increasing value of the latter factor induces the variation of the density of predator species. The simulation results presented in this work reveal the inherent mechanism of our model. Moreover, on the basis of our analysis, the slight value changes in d1 lead to density changes of the predator. This observation shows that the diffusive rate of the prey has a great effect on prey and the whole system. Above all, the intrinsic agents, i.e., diffusion and predation, control the persistence or extinction of the spatial predator-prey model [45].However, it is noticeable that there is a little defect in the analysis of pattern selection via amplitude equations. That is, pattern selection is only valid when the value of the controlled parameterd1 is set close to the threshold d1T. Thus, pattern selection is only valid when the value is very close to the controlled parameter d1T. This inspires us to think about the best distance of the parameter away from the controlled parameter. Besides, the method of the amplitude equations is also applicable in the epidemic model and the vegetation model to study the pattern formation of the epidemic and vegetation in the future [46–49].



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*Source: 6657405-2021-03-11.xml*