An Euler difference scheme for a three-dimensional predator-prey model is considered and we introduce a new approach to show the global stability of the scheme. For this purpose, we partition the three-dimensional space and calculate the sign of the rate change of population of species in each partitioned region. Our method is independent of dimension and then can be applicable to other dimensional discrete models. Numerical examples are presented to verify the results in this paper.
1. Introduction
Vito Volterra proposed a differential equation model to explain the observed increase in predator fish and corresponding decrease in prey fish in 1926. At the same time, the equations studied by Volterra were derived independently by Alfred Lotka (1925) to describe a chemical reaction. Many predator-prey models have been studied and a classic predator-prey model is given by(1)dxdt=xr1-a11x-a12y,dydt=y-r2+a21x-a22y,where ri>0, aij>0, and x and y denote the population sizes of the prey and predator, respectively.
It is recognized that the rate of prey capture per predator cannot increase indefinitely as the number of prey increases. Instead, the rate of prey capture is saturated when the population of prey is relatively large. Then such nonlinear functional responses are employed to describe the phenomena of predation including the Holling types [1–5], Beddington-DeAngelis type [6–8], Crowley-Martin type [9–11], and Ivlev type of functional response [12–14].
In particular, similar phenomena are observed in the interactions in chemical reactions and molecular events when one species is abundant. Thus linear response function, Michaelis-Menten kinetics, and Hill function are related to Holling types I, II, and III, respectively. Holling type IV is also called the Monod-Haldane function [4, 5].
The functional responses have been applied to predator-prey models to express the Allee effect [15–19], which describes a positive relation between population density and the per capita growth rate.
Most of researches on the predator-prey models assume that the distribution of the predators and prey is homogeneous, which leads to ordinary differential equations. However both predators and prey have the natural tendency to diffuse, so that there have been models to take into account the inhomogeneous distribution of the predators and prey [20–22].
On the other hand, population is inevitably affected by environmental noise in nature. Therefore, many authors have taken stochastic perturbation into deterministic models [23–25].
There are a number of works investigating continuous time predator-prey models, but relatively few theoretical papers are published on their discretized models [26, 27]. As far as we know, there is no theoretical research on the global stability of the discrete-time models of type (1) with more than two species except for [28]. The author in [28] introduced a method to present global stability for the case that all species coexist at a unique equilibrium. Then a new approach needs to be developed for the other cases.
For explaining our new approach, we consider a model with one prey and two predators:(2)dxidt=xiσiri+∑1≤j≤i-1aijxj-∑i≤j≤3aijxj,where i=1,2,3, σ1=1, σ2=σ3=-1, and(3)r3a31<r1a11<r2a21,(4)r1a12<r3a32,(5)a21a23<a31a33.Here x1 denotes the population number of the prey; x2 and x3 denote the population numbers of the predators. Letting (6)fix1,x2,x3=σiri+∑1≤j≤i-1aijxj-∑i≤j≤3aijxj,the Euler difference scheme for (2) is as follows:(7)ΔxkiΔt=xkifixk1,xk2,xk3,k=0,1,…,where x0i>0(i=1,2,3), Δxki=xk+1i-xki, and Δt is a step size.
The three conditions (3)–(5) mean that the two planes f1(x,y,z)=0 and f3(x,y,z)=0 intersect in the first octant, and the plane f2(x,y,z)=0 is not intersected with either f1(x,y,z)=0 or f3(x,y,z)=0 in the first octant. For example, let (ri,ai1,ai2,ai3)(1≤i≤3) be (8)2,1,2,2,3,1,1,1,2,2,1,1,respectively. Then the three conditions (3)–(5) are satisfied and Figure 1 shows the three planes and regions with the globally stable point of scheme (7). The global stability of the point will be shown in Section 3.
Three planes and regions with the globally stable point. (a) The three planes fi(x,y,z)=0(1≤i≤3) in (0,4)×(0,3)×(0,2). The magenta circle denotes the globally stable point (6/5,0,2/5) of scheme (7). (b) Five regions projected on the xy-plane. (c) Five regions projected on the xz-plane.
Using Theorem 4.1 in [28], we have the positivity and boundedness of the solutions of scheme (7). Letting χi(1≤i≤3) satisfy(9)χi<1+σiriΔt-∑i+1≤j≤naijχjΔt2aiiΔt,σiri+∑1≤j≤i-1aijχjaii<χi,σiri+∑1≤j≤i-1aijχjΔt<1,we have that for all k(10)xk1,xk2,xk3∈0,χ1×0,χ2×0,χ3.For small values of Δt, there exist infinitely many χi(1≤i≤3) satisfying (9).
Consider the five regions(11)I=x,y,z∈S∣f1x,y,z≥0,f2x,y,z<0,f3x,y,z≤0,II=x,y,z∈S∣f1x,y,z≥0,f2x,y,z<0,f3x,y,z≥0,III=x,y,z∈S∣f1x,y,z≤0,f2x,y,z<0,f3x,y,z≤0,IV=x,y,z∈S∣f1x,y,z≤0,f2x,y,z≤0,f3x,y,z≥0,V=x,y,z∈S∣f1x,y,z<0,f2x,y,z≥0,f3x,y,z>0,where S={(x,y,z)∈R3∣x>0,y>0,z>0}∩∏1≤i≤3(0,χi) (see Figure 1).
For convenience we denote the set I by (12)I+,-,-or +,-,-,and then(13)P∈I+,-,-iff f1P≥0,f2P<0,f3P≤0.We adapt similar notations for the other regions II to V.
Note that region I has the property(14)supx∣x,y,z∈I+,-,-≤r1a11,since f1(x,y,z)=r1-a11x-a12y-a13z≥0 for all (x,y,z)∈I(+,-,-).
2. Method
In this section we present theorems that describe our approach. The theorems will be used to obtain the global stability of scheme (7) in the next section.
Assume that Δt satisfies(15)Δtχ1a11+a13a21a23+χ2a22+χ3a33+a32a13a12<1,(16)Δt1-a33χ3Δta31χ12<a31r2-a21r3a21.Let P=(P1,P2,P3) and T=(T1,T2,T3) be the vector function defined on S by (17)TiP=Pi1+ΔtfiP,i=1,2,3.Then scheme (7) can be written as (18)xk+11,xk+12,xk+13=Txk1,xk2,xk3.Note that the map T has the three fixed points with all nonnegative components, (0,0,0), r1/a11,0,0, and ϑ=(ϑ1,0,ϑ3), satisfying(19)f1ϑ=f3ϑ=0.Then the fixed point ϑ with ϑ1=r1a33+r3a13/a11a33+a31a13 and ϑ3=r1a31-r3a11/a13a31+a33a11 is locally stable as follows.
Lemma 1.
Let ϑ be the fixed point of the map T satisfying (19). Assume that Δt satisfies (15). Then the fixed point ϑ is asymptotically stable.
Proof.
Applying the linearization method to the discrete system of (7) at ϑ, the matrix of the linearized system has the three eigenvalues (20)1+f2ϑΔt,1-12ϑ1a11+ϑ3a33Δt±12ϑ1a11-ϑ3a332-4ϑ1ϑ3a13a311/2Δt.Substituting ϑ1=(r1a33+r3a13)(a11a33+a31a13)-1 to (21)f2ϑ=-r2+a21ϑ1-a23ϑ3<-r2+a21ϑ1,and using (3), we obtain (22)f2ϑ<0.In addition, (15) gives that the other two eigenvalues have magnitudes less than 1. Hence the spectral radius of the matrix is less than 1, which completes the proof.
The two fixed points (0,0,0) and r1/a11,0,0 are unstable since the matrices of the linearized system at (0,0,0) and r1/a11,0,0 have the eigenvalues 1+f1(0,0,0)Δt and 1+f3(r1/a11,0,0)Δt with f1(0,0,0)>0 and f3(r1/a11,0,0)>0, respectively. Then we only consider the fixed point ϑ=(ϑ1,0,ϑ3) of the map T to show global stability in the next section.
Our methodology to obtain the global stability is based on the approach to determine regions among regions I to V in which T(P) is contained by calculating the sign of fiT(P) for P in each region. Then we use the sign symbol P∈(+,∗,∗) as follows.
P∈(+,∗,∗) if and only if f1(P)≥0 and the signs of fi(P)(i=2,3) are unknown.
Other sign symbols are similarly defined.
Theorem 2.
Suppose that Δt satisfies (9) and (15). Let s∈{+,-}:(23)IfP∈s,-s,-s,thenthesignoff1TPiss.
Proof.
Let P=(P1,P2,P3). Using the definition of f1, we have(24)f1TP=r1-a11T1P-a12T2P-a13T3P=r1-∑1≤i≤3a1iPi1+ΔtfiP=r1-∑1≤i≤3a1iPi-∑1≤i≤3Δta1iPifiP=f1Pi-∑1≤i≤3Δta1iPifiP=1-Δta11P1f1P-Δta12P2f2P+a13P3f3P.Note that if f2(x,y,z)=0 for nonnegative x,y, and z, then (25)x=1a21r2+a22y+a23z,with which the nonnegativity of x,y, and z gives (26)f1x,y,z=r1-a11x-a12y-a13z=r1-a111a21r2+a22y+a23z-a12y-a13z≤r1-a111a21r2=a11r1a11-r2a21<0,and similarly (27)f3x,y,z>0,due to (3). Hence there exist no nonnegative numbers x,y, and z such that f1(x,y,z)=f2(x,y,z)=f3(x,y,z)=0. Since P∈(s,-s,-s) and it is impossible that f1(P)=f2(P)=f3(P)=0, (24) gives that the sign of f1(T(P)) is s.
Using Theorem 2, we have the following:(28)If P∈I+,-,-,thenTP∉III∪IV∪V-,∗,∗.(29)If P∈V-,+,+,then TP∉I+,-,-.
Remark 3.
Suppose that Δt satisfies (9) and (15). Let s∈{+,-}. Then the definitions of f2 and f3 yield(30)f2TP=1-Δta22P2f2P+Δta21P1f1P-a23P3f3P,(31)f3TP=1-Δta33P3f3P+Δta31P1f1P+a32P2f2P,and hence we can also obtain the property as in Theorem 2.
If P∈(s,s,-s), then the sign of f2(T(P)) is s.
If P∈(s,s,s), then the sign of f3(T(P)) is s.
Using (a) and (b), we can obtain the following: (32)If P∈IV-,-,+,then TP∉V-,+,+,(33)If P∈III-,-,-,then TP∉II∪IV∪V∗,∗,+.
It follows from (28) and (29) that every point P in a region cannot move by the map T to regions with three different signs. In the case of regions with two different signs, it is also impossible by the following theorem.
Theorem 4.
Suppose that Δt satisfies (9), (15), and (16).
If P∈II(+,-,+), then T(P)∉V(-,+,+).
If P∈V(-,+,+), then T(P)∉II(+,-,+).
If P∈II(+,-,+), then T(P)∉III(-,-,-).
If P∈III(-,-,-), then T(P)∉II(+,-,+).
If P∈III(-,-,-), then T(P)∉V(-,+,+).
If P∈V(-,+,+), then T(P)∉III(-,-,-).
Proof.
(a) Suppose that T(P)∈V(-,+,+). Then (24) and (30) with P∈II(+,-,+) give(34)1-Δta11P1f1P+Δta12P2f2P<Δta13P3f3P,(35)a23P3f3P≤a21P1f1P,respectively. Combining (34) and (35), we obtain(36)1-Δta11P1f1P<Δta13a21P1a23f1P.Since f1(P)≥0, inequality (36) is a contradiction to (15). Therefore the proof of (a) is completed.
(b) Suppose that T(P)∈II(+,-,+). Then the inclusion of P∈V(-,+,+) gives a contradiction: (37)0≤f1TP=1-Δta11P1f1P-Δta12P2f2P+a13P3f3P<0.
(c) Suppose that T(P)∈III(-,-,-). Then (24) and (31) with P∈II(+,-,+) give (38)1-Δta11P1f1P+Δta12P2f2P≤Δta13P3f3P,1-Δta33P3f3P≤Δta32P2f2P,which yield (39)1-Δta33P3f3P≤Δta32a13P3a12f3P.Hence, if f3(P)>0 or f3(P)=0, then(40)1-Δta33P3<Δta32a13P3a12or fiP=01≤i≤3,respectively; these are contradictions due to (15) and the fact that there is no solution of the system of equations fi(x,y,z)=0(i=1,2,3).
(d) and (e) are proved by (33).
(f) Suppose that T(P)∈III(-,-,-). Then (31) with P∈V(-,+,+) gives(41)0<f3P≤Δt1-Δta33P3a31P1f1P.It follows from f2(P)=-r2+a21P1-a22P2-a23P3≥0, (5), and (3) that(42)f3P≥-r3+a311a21r2+a22P2+a23P3+a32P2-a33P3>1a21-a21r3+a31r2+a31a23-a21a33P3>1a21-a21r3+a31r2.Since -a21r3+a31r2>0 by (3), inequality (41) with (42) is a contradiction to (16), which completes the proof.
Theorem 5.
Suppose that Δt satisfies (9). If P∈V, then Tm1(P)∉V for some m1.
Proof.
Suppose that Tm(P)∈V(-,+,+) for all m. Then there exist constants x∗,y∗, and z∗ such that (43)x∗=limk→∞xk1≥0,y∗=limk→∞xk2>0,z∗=limk→∞xk3>0,and so(44)f2x∗,y∗,z∗=f3x∗,y∗,z∗=0.This is a contradiction since the system of (44) gives that(45)0<a22a31+a32a21y∗=-a31r2-a21r3-a23a31-a33a21z∗<0,due to both (3) and (4) with z∗>0.
The results we obtained are summarized in Table 1.
The regions containing T(P). The symbols I to V denote the regions defined in (11). The region I in the second column denotes P∈I, the symbol × at (P,T(P))=(I,III) means T(P)∉III, and then the two circles ∘ together at both (I, I) and (I, II) denote that T(P)∈I∪II by (28). The symbol ✗ denotes that P∈ V and Tm1(P)∉ V for some m1 by Theorem 5.
T(P)
Equation and theorem
I
II
III
IV
V
P
I
∘
∘
×
×
×
Equation (28)
II
∘
∘
×
∘
×
Theorem 4(c) and (a)
III
∘
×
∘
×
×
Equation (33)
IV
∘
∘
∘
∘
×
Equation (32)
V
×
×
×
∘
✗
Equation (29); Theorem 4(b) and (f); and Theorem 5
Finally, using Table 1, we can obtain the following theorem.
Theorem 6.
Suppose that Δt satisfies (9), (15), and (16). If P∈I∪II∪III∪IV∪V, then for all sufficiently large m(46)TmP∈I∪II∪III∪IV∗,-,∗.
Remark 7.
Table 1 and Theorem 6 are obtained for the case that only two planes fi(x,y,z)=0(i=1,3) intersect in the first octant. We can also apply the approach to the other cases and then have a table and a theorem similar to Table 1 and Theorem 6.
3. Global Stability
In this section we show that the fixed point ϑ=(ϑ1,0,ϑ3) of the map T satisfying (19) is globally stable. Let Δt satisfy that (47)ai1χ1+ϑ1+ai2χ2+ai3χ3+ϑ3Δt<12,i=1,3,(48)Δt<mina31a11,a13a332a31a12+Aϑ1M+2a13a32+Aϑ3M,where A=3∑1≤j≤3(a1j+a3j) and M=6max1≤i,j≤3aij.
Theorem 8.
Let ϑ be the point defined in (19). Suppose that Δt satisfies (9), (15), (16), (47), and (48). Assume that the initial point (x01,x02,x03) of the Euler scheme (7) satisfies (49)x01,x02,x03∈∏1≤i≤30,χi.Then (50)limk→∞xk1,xk2,xk3=ϑ.
Proof.
Theorem 6 gives that for all k(51)xk1,xk2,xk3∈I∪II∪III∪IV∗,-,∗,and then (10) gives that for a nonnegative constant y∗:(52)y∗=limk→∞xk2.
Now we claim that y∗=0: suppose, on the contrary, that(53)y∗>0.
Applying both (52) and (53) to (7), we have (54)limk→∞f2xk1,xk2,xk3=0,which implies that there exists a constant ϵ0 such that for all sufficiently large k(55)0<ϵ0<a21maxr2a21-r1a11,r2a21-r3a31,(56)f2xk1,xk2,xk3>-ϵ0.Then (56) and (55) give xk1>1/a21(r2-ϵ0)>r1/a11, and so (14) gives(57)xk1,xk2,xk3∉I+,-,-.Using (56), we have (58)xk1>1a21r2-ϵ0+a22xk2+a23xk3,and so it follows from (42) and (55) that for all sufficiently large k(59)f3xk1,xk2,xk3>1a21a31r2-ϵ0-a21r3>0,which gives(60)xk1,xk2,xk3∉III-,-,-.Hence (51) with (57) and (60) gives that for all sufficiently large k(61)xk1,xk2,xk3∈II∪IV∗,-,+.Both (61) and (53) yield that there exist the two limits: (62)x∗=limk→∞xk1,z∗=limk→∞xk3>0,and then (63)f2x∗,y∗,z∗=f3x∗,y∗,z∗=0.This is a contradiction due to (45) and finally we obtain the claim (64)limk→∞xk2=0.Consider the function Vk defined by(65)Vk=a31xk1-ϑ1lnxk1+a13xk3-ϑ3lnxk3.Letting a^ij=aijΔt and (66)ϑ1k=ϑ1-xk1,ϑ3k=ϑ3-xk3,scheme (7) and the fixed point (19) yield (67)Δxk1=xk1a^11ϑ1k-a^12xk2+a^13ϑ3k,Δxk3=xk3-a^31ϑ1k+a^32xk2+a^33ϑ3k.Then the mean value theorem gives that for some α,β with 0<α, β<1(68)Vk+1-Vk=a31Δxk11-ϑ1Δlnxk1Δxk1+a13Δxk31-ϑ3Δlnxk3Δxk3=a31a^11ϑ1k-a^12xk2+a^13ϑ3k·xk1-ϑ1xk1αΔxk1+xk1+a13-a^31ϑ1k+a^32xk2+a^33ϑ3k·xk3-ϑ3xk3βΔxk3+xk3.Note that (69)xk1αΔxk1+xk1=1-αa11ϑ1k-a12xk2+a13ϑ3kΔtαa11ϑ1k-a12xk2+a13ϑ3kΔt+1≡1-C1ϑ1k+C2xk2+C3ϑ3kΔt,xk3βΔxk3+xk3=1-β-a31ϑ1k+a32xk2+a33ϑ3kΔtβ-a31ϑ1k+a32xk2+a33ϑ3kΔt+1≡1-C4ϑ1k+C5xk2+C6ϑ3kΔt,where (47) gives(70)max1≤i≤6Ci<2max1≤i,j≤3aij.
Now suppose, on the contrary, that (xk1,xk2,xk3) does not converge to (ϑ1,0,ϑ3).
Since ϑ is asymptotically stable by Lemma 1, the supposition with limk→∞xk2=0 implies that(71)ϑ1k+ϑ3k has a positive lower bound.Again, using limk→∞xk2=0 with (71), we can have that for all sufficiently large k(72)xk2<Δtϑ1k+ϑ3k.Then (68) becomes(73)Vk+1-Vk≤-Δta31a11-C7Δtϑ1k2-Δta13a33-C8Δtϑ3k2,where (74)maxC7,C8<2a31a12+Aϑ1C123+2a13a32+Aϑ3C456,with A=3∑1≤j≤3(a1j+a3j) and |Cijk|=|Ci|+|Cj|+|Ck|. Hence (73) with (48) becomes (75)Vk+1-Vk≤-C9ϑ1k2+ϑ3k2Δt,for a positive constant C9, so that there exists a positive constant C such that for all sufficiently large k(76)Vk+1-Vk≤-CΔt,and then (77)Vk≤V0-kCΔt.Hence we have (78)limk→∞Vk=limk→∞V0-kCΔt=-∞,which is a contradiction due to (10).
Remark 9.
The global stability in Theorem 8 is obtained for the discrete predator-prey model with one prey and two predators by using both Theorem 6 and the Lyapunov type function Vk. At first, we show that one predator is extinct by Theorem 6 and then we can apply the function Vk which was used in the lower dimensional case: the two-dimensional discrete model with one prey and one predator. Therefore our approach can utilize the methods used in lower dimensional models.
On the other hand, in the case that (79)r1a11<r3a31<r2a21,r1a12<r3a32,a21a23<a31a33,region II is empty and similarly we can obtain the global stability of the fixed point r1/a11,0,0 of the map T by using Theorems 2–5 without applying a Lyapunov type function like Vk.
4. Numerical Examples
In this section, we consider the Euler difference scheme for (2):(80)Δxk1=xk12-xk1-2xk2-2xk3Δt,Δxk2=xk2-3+xk1-xk2-xk3Δt,Δxk3=xk3-2+2xk1+xk2-xk3Δt,where Δt=0.001 and k=0,1,…. Applying Theorem 4.1 in [28] to (80), we can obtain the positivity and boundedness of the solutions. For example,(81)xk1,xk2,xk3∈0,4×0,2×0,8∀k.The fixed point ϑ in (19) becomes ϑ=(6/5,0,2/5) which is denoted by the magenta circle in Figures 1 and 2, and the value Δt=0.001 satisfies all conditions (9), (15), (16), (47), and (48). Consequently, ϑ=(6/5,0,2/5) is globally stable, which is demonstrated in Figure 2 for five different initial points (x01,x02,x03) contained in regions I to V, respectively.
Trajectories for five different initial points denoted by the black circles. The initial points are (0.5, 0.375, 0.3562), (1.15, 0.2125, 0.1281), (0.125, 1.5, 1.5), (1.5, 0.125, 0.2813), and (4, 0.5, 0.25), which are contained in regions I, II, III, IV, and V defined in (11), respectively. The brown dotted lines mean the trajectories which converge to ϑ=(6/5,0,2/5), denoted by the magenta circle. The three planes fi=0 are depicted in Figure 1.
In order to verify the results in Table 1, we mark the regions containing T(P) for P located in the five trajectories in Figure 2 and present the result in Figure 3, which demonstrates that the regions containing T(P) follow the rule in Table 1. For example, we can see in Figure 3 that T(P) cannot be contained in V for every P∈I∪II∪III∪IV, and if P∈II, then T(P) can be contained only in II∪IV.
The region containing (xk1,xk2,xk3) at time t=kΔt for the five trajectories in Figure 2. (a) The initial point (x01,x02,x03) is (0.5, 0.375, 0.3562), denoted by the black circle and located at (0,I). The green segments denote the regions containing (xk1,xk2,xk3). The region having P is connected to the region having T(P) by the brown line. The other figures (b), (c), (d), and (e) are similarly obtained by using the different initial points in Figure 2.
5. Conclusion
In this paper, we have developed a new approach to obtain the global stability of the fixed point of a discrete predator-prey system with one prey and two predators. The main idea of our approach is to describe how to trace the trajectories. In this process, we calculate the sign of the rate change of population of species, so that we call our method the sign method. Although we have applied our sign method for the three-dimensional discrete model, the sign method can be utilized for two-dimensional and other higher dimensional discrete models.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the 2015 Research Fund of University of Ulsan.
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