A New Approach to Global Stability of Discrete Lotka-Volterra Predator-Prey Models

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.


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 where   > 0,   > 0, 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][2][3][4][5], Beddington-DeAngelis type [6][7][8], Crowley-Martin type [9][10][11], and Ivlev type of functional response [12][13][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 predatorprey models to express the Allee effect [15][16][17][18][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][21][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][24][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  = 1, 2, 3, Here  1 denotes the population number of the prey;  2 and  3 denote the population numbers of the predators.Letting the Euler difference scheme for ( 2) is as follows: where   0 > 0 ( = 1, 2, 3), Δ   =   +1 −    , and Δ is a step size.

Method
In this section we present theorems that describe our approach.The theorems will be used to obtain the global stability of scheme (7) The magenta circle denotes the globally stable point (6/5, 0, 2/5) of scheme (7).(b) Five regions projected on the -plane.(c) Five regions projected on the -plane.
Proof.Applying the linearization method to the discrete system of ( 7) at , the matrix of the linearized system has the three eigenvalues Substituting 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.
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.
Other sign symbols are similarly defined.
Using Theorem 2, we have the following: Remark 3. Suppose that Δ satisfies ( 9) and (15).Let  ∈ {+, −}.Then the definitions of  2 and  3 yield and hence we can also obtain the property as in Theorem 2.
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.
The results we obtained are summarized in Table 1.
Finally, using Table 1, we can obtain the following theorem.
Table 1: The regions containing ().The symbols I to V denote the regions defined in (11).The region I in the second column denotes  ∈ I, the symbol × at (, ()) = (I, III) means () ∉ III, and then the two circles ∘ together at both (I, I) and (I, II) denote that () ∈ I ∪ II by (28).The symbol M denotes that  ∈ V and   1 () ∉ V for some  1 by Theorem 5.
Theorem 8. Let  be the point defined in (19).Suppose that Δ satisfies ( 9), ( 15), ( 16), (47), and (48).Assume that the initial point ( 1 0 ,  2 0 ,  3 0 ) of the Euler scheme (7) satisfies Proof.Theorem 6 gives that for all and then (10) gives that for a nonnegative constant  * : Now we claim that  * = 0: suppose, on the contrary, that Applying both ( 52) and ( 53) to ( 7), we have lim which implies that there exists a constant  0 such that for all sufficiently large Then ( 56) and (55) give  1  > (1/ 21 )( 2 −  0 ) >  1 / 11 , and so (14) gives Using (56), we have and so it follows from (42) and ( 55) that for all sufficiently large which gives Hence (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 lim Consider the function   defined by Letting â =   Δ and scheme (7) and the fixed point (19) yield Then the mean value theorem gives that for some ,  with 0 < ,  < 1 Note that for a positive constant  9 , so that there exists a positive constant C such that for all sufficiently large and then Hence we have lim 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

Conclusion
In this paper, we have developed a new approach to obtain the global stability of the fixed point of a discrete predatorprey 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.