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Stability and approximate analytic solutions of the fractional Lotka-Volterra equations for three competitors
- Changhong Guo^{1} and
- Shaomei Fang^{2}Email author
https://doi.org/10.1186/s13662-016-0943-y
© Guo and Fang 2016
- Received: 26 March 2016
- Accepted: 16 August 2016
- Published: 25 August 2016
Abstract
This paper studies the fractional Lotka-Volterra equations for three competitors, since the fractional derivatives possess the properties of good memory and have great biological significance. First of all, the equilibrium points and asymptotic stability for the equations are studied by the stability analysis method. As expected, the fractional-order differential equations are, at least, as stable as their integer-order counterpart. Second, some approximate analytic solutions for this systems are obtained by the stability analysis method and the homotopy perturbation method, which are expressed in the form of the Mittag-Leffler function. The results show that it takes less time for the population to get close to the equilibrium point as the time derivatives increase. Comparing with the classical ones, the fractional Lotka-Volterra equations, no matter whether the fractional derivatives have big or small order, if both take more time, even multiplied time, for the system to reach the equilibrium point, then that may explain the memory properties. Furthermore, some numerical analyses are carried out and verify the theoretical analysis.
Keywords
- fractional Lotka-Volterra equations
- asymptotic stability
- approximate analytic solutions
- homotopy perturbation method
1 Introduction
Since the classical Lotka-Volterra model has been applied to all kinds of problem in population biology, neural networks, and others, this model has been studied by many ecologists and mathematicians. For now, many important and interesting results have been found, such as the existence and uniqueness of solutions [4, 5], global asymptotic stability [6], Hopf bifurcation [7], and extinction [8]. For some other research results, see [9, 10] and the references therein.
The rest of paper is organized as follows. In Section 2, we briefly give some preliminaries. In Section 3, we will study the equilibrium points, asymptotic stability, and the approximate analytic solutions for the fractional Lotka-Volterra system (9)-(12) with the help of a stability analysis. In Section 4, other forms of approximate analytic solutions are studied by the homotopy perturbation method. In Section 5, some numerical results are presented to verify the theoretical analysis. In Section 6, we draw some conclusions.
2 Some preliminaries
In this section, we will briefly give some preliminaries of the fractional calculus and the homotopy perturbation method, which will be used in the next sections.
2.1 Fractional calculus
For the properties of the fractional integrals and derivatives, we have the following results [11].
Lemma 1
Lemma 2
Proof
These results can be proved easily with the help of the Laplace transform and the properties of the Mittag-Leffler function. We omit the details for simplicity. □
Remark
The Mittag-Leffler function (23) was in fact introduced by Agarwal [26] and is the two-parameter generalization of the exponential function \(e^{z}\), which plays a very important role in the theory of population biology. The exponential function can be found in the classical Malthus model and logistic model, and thus the Mittag-Leffler function is more generalized to describe a real-world growing population.
Lemma 3
2.2 Homotopy perturbation method
3 Equilibrium points and their approximate solutions
Finally, combining (33), (42), and (45)-(47), we can obtain the solutions for the system (9)-(12).
Furthermore, using the result of Matignon [27] then if the eigenvalues of the matrix A are negative (\(|\operatorname{arg}(\lambda_{1})|>\alpha\pi /2\), \(|\operatorname{arg}(\lambda_{2})|>\beta\pi/2\), \(|\operatorname{arg}(\lambda_{3})|>\gamma\pi/2\)), the equilibrium point \((x^{\mathrm{eq}}, y^{\mathrm{eq}}, z^{\mathrm{eq}})\) is locally asymptotically stable. This result confirms the statement that fractional-order differential equations are, at least, as stable as their integer-order counterpart [14].
At last, according to asymptotic stability results, the equilibrium point \((x^{\mathrm{eq}}, y^{\mathrm{eq}}, z^{\mathrm{eq}})= (\frac {1}{a+b+1}, \frac{1}{a+b+1}, \frac{1}{a+b+1} )\) is locally asymptotically stable if \(0< a+b<2\). That is to say, if the competition affects the coefficients among the three populations a and b satisfying the condition \(0< a+b<2\), then the densities of the populations X, Y, and Z will eventually trend to be stable and close to the equilibrium point \((x^{\mathrm{eq}}, y^{\mathrm{eq}}, z^{\mathrm{eq}})\). The numerical simulations in the following section will also support this result.
4 Approximate solutions by HPM
5 Numerical results and discussion
In the previous sections, we have obtained the approximate analytic solutions as found in (51)-(53) and (68) for the fractional Lotka-Volterra equations (9)-(12) by the stability analysis and HPM. Thus in this section, we present some numerical simulation. For simplicity, we take \(a=b=0.5\), which means the competition affecting coefficients among the three populations are the same, and we choose the initial values \(x_{0}=1\), \(y_{0}=0.8\), \(z_{0}=0.2\). Thus according to the asymptotic stability results, we know the positive equilibrium point is \((x^{\mathrm{eq}}, y^{\mathrm{eq}}, z^{\mathrm{eq}})=(\frac{1}{2}, \frac{1}{2}, \frac {1}{2})\), and this equilibrium point is locally asymptotically stable.
6 Conclusions
Since natural biological systems have memory properties, fractional differential equations provide an excellent instrument in this respect in comparison with the classical integer-order counterparts. Thus in this paper, we studied the fractional Lotka-Volterra equations for three competitors under some symmetry assumptions. First we studied the equilibrium points and asymptotic stability for the equations by the stability analysis method, and we argued that the fractional-order differential equations are, at least, as stable as their integer-order counterpart. Second, we applied the stability analysis method and the homotopy perturbation method to obtain the approximate analytic solutions for the fractional systems, which are expressed in the form of the Mittag-Leffler function. As the numerical results also show, it takes less time for the population to get close to the equilibrium point as the time derivatives increases. However, compared with the classical Lotka-Volterra equations, we can see that the fractional Lotka-Volterra equations, no matter whether the order of the fractional derivatives is big or small, if both take more time, even multiplied for the system to reach the equilibrium, that may explain the memory properties and have great biological significance.
Declarations
Acknowledgements
This research was supported by the National Natural Science Foundation of China (Nos. 11426069, 11271141), and the Starting Foundation for Doctors of Guangdong University of Technology (No. 15ZK0009).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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