Abstract
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 bywhere , , and and 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 , , and , respectively. Holling type 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:where , , , andHere denotes the population number of the prey; and denote the population numbers of the predators. Letting the Euler difference scheme for (2) is as follows:where , , and is a step size.
The three conditions (3)–(5) mean that the two planes and intersect in the first octant, and the plane is not intersected with either or in the first octant. For example, let be 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.
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Using Theorem 4.1 in [28], we have the positivity and boundedness of the solutions of scheme (7). Letting satisfywe have that for all For small values of , there exist infinitely many satisfying (9).
Consider the five regionswhere (see Figure 1).
For convenience we denote the set by and thenWe adapt similar notations for the other regions to .
Note that region has the propertysince for all .
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 satisfiesLet and be the vector function defined on by Then scheme (7) can be written as Note that the map has the three fixed points with all nonnegative components, , , and , satisfyingThen the fixed point with and is locally stable as follows.
Lemma 1. Let be the fixed point of the map satisfying (19). Assume that satisfies (15). Then the fixed point is asymptotically stable.
Proof. Applying the linearization method to the discrete system of (7) at , the matrix of the linearized system has the three eigenvalues Substituting to and using (3), we obtain 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 and are unstable since the matrices of the linearized system at and have the eigenvalues and with and , respectively. Then we only consider the fixed point of the map 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 is contained by calculating the sign of for in each region. Then we use the sign symbol as follows.
if and only if and the signs of are unknown.
Other sign symbols are similarly defined.
Theorem 2. Suppose that satisfies (9) and (15). Let :
Proof. Let . Using the definition of , we haveNote that if for nonnegative , and , then with which the nonnegativity of , and gives and similarly due to (3). Hence there exist no nonnegative numbers , and such that . Since and it is impossible that , (24) gives that the sign of is .
Using Theorem 2, we have the following:
Remark 3. Suppose that satisfies (9) and (15). Let . Then the definitions of and yieldand hence we can also obtain the property as in Theorem 2.(a)If , then the sign of is .(b)If , then the sign of is .Using (a) and (b), we can obtain the following:
It follows from (28) and (29) that every point in a region cannot move by the map 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 satisfies (9), (15), and (16).(a)If , then .(b)If , then .(c)If , then .(d)If , then .(e)If , then .(f)If , then .
Proof. (a) Suppose that . Then (24) and (30) with giverespectively. Combining (34) and (35), we obtainSince , inequality (36) is a contradiction to (15). Therefore the proof of (a) is completed.
(b) Suppose that . Then the inclusion of gives a contradiction: (c) Suppose that . Then (24) and (31) with give which yield Hence, if or , thenrespectively; these are contradictions due to (15) and the fact that there is no solution of the system of equations .
(d) and (e) are proved by (33).
(f) Suppose that . Then (31) with givesIt follows from , (5), and (3) thatSince by (3), inequality (41) with (42) is a contradiction to (16), which completes the proof.
Theorem 5. Suppose that satisfies (9). If , then for some .
Proof. Suppose that for all . Then there exist constants , and such that and soThis is a contradiction since the system of (44) gives thatdue to both (3) and (4) with .
The results we obtained are summarized in Table 1.
Finally, using Table 1, we can obtain the following theorem.
Theorem 6. Suppose that satisfies (9), (15), and (16). If , then for all sufficiently large
Remark 7. Table 1 and Theorem 6 are obtained for the case that only two planes 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 of the map satisfying (19) is globally stable. Let satisfy that where and .
Theorem 8. Let be the point defined in (19). Suppose that satisfies (9), (15), (16), (47), and (48). Assume that the initial point of the Euler scheme (7) satisfies Then
Proof. Theorem 6 gives that for all and then (10) gives that for a nonnegative constant :Now we claim that : suppose, on the contrary, thatApplying both (52) and (53) to (7), we have which implies that there exists a constant such that for all sufficiently large Then (56) and (55) give , and so (14) givesUsing (56), we have and so it follows from (42) and (55) that for all sufficiently large which givesHence (51) with (57) and (60) gives that for all sufficiently large Both (61) and (53) yield that there exist the two limits: and then This is a contradiction due to (45) and finally we obtain the claim Consider the function defined byLetting and scheme (7) and the fixed point (19) yield Then the mean value theorem gives that for some with , Note that where (47) givesNow suppose, on the contrary, that does not converge to .
Since is asymptotically stable by Lemma 1, the supposition with implies thatAgain, using with (71), we can have that for all sufficiently large Then (68) becomeswhere with and . Hence (73) with (48) becomes for a positive constant , so that there exists a positive constant such that for all sufficiently large and then Hence we have 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 . At first, we show that one predator is extinct by Theorem 6 and then we can apply the function 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 region is empty and similarly we can obtain the global stability of the fixed point of the map by using Theorems 2–5 without applying a Lyapunov type function like .
4. Numerical Examples
In this section, we consider the Euler difference scheme for (2):where and . Applying Theorem 4.1 in [28] to (80), we can obtain the positivity and boundedness of the solutions. For example,The fixed point in (19) becomes which is denoted by the magenta circle in Figures 1 and 2, and the value satisfies all conditions (9), (15), (16), (47), and (48). Consequently, is globally stable, which is demonstrated in Figure 2 for five different initial points contained in regions to , respectively.
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In order to verify the results in Table 1, we mark the regions containing for located in the five trajectories in Figure 2 and present the result in Figure 3, which demonstrates that the regions containing follow the rule in Table 1. For example, we can see in Figure 3 that cannot be contained in for every , and if , then can be contained only in .
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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.