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- Open Access
Permanence and extinction of a high-dimensional stochastic resource competition model with noise
- Li Wang^{1},
- Xiaoqiang Wang^{2} and
- Qimin Zhang^{1}Email author
https://doi.org/10.1186/s13662-018-1891-5
© The Author(s) 2018
- Received: 16 May 2018
- Accepted: 15 November 2018
- Published: 29 November 2018
Abstract
In this paper, we investigate the asymptotic behavior for a kind of resource competition model with environmental noises. Considering the impact of white noise on birth rate and death rate separately, we first prove the existence of a positive solution, and then a sufficient condition to maintain permanence and extinction is obtained by using a proper Lyapunov functional, stochastic comparison theorem, strong law of large numbers for martingales, and several important inequalities. Furthermore, the stochastic final boundedness and path estimation are studied. Finally, the fact that the intensity of white noise has a very important influence on the permanence and extinction of the system’s solution is illustrated by some numerical examples.
Keywords
- Permanence and extinction
- Lyapunov functional
- Stochastic comparison theorem
- Strong number theorem of martingale
1 Introduction
As we all know, the classical Lotka–Volterra model can well describe the competition among different populations, thus it has been one of the most important models in the field of mathematical ecology. In recent decades, it was found that the Lotka–Volterra model can do nothing about forecasting except portraying the densities of the interactive population, thus also cannot describe the competitive mechanism. The Lotka–Volterra model can only do feedback estimation by the result of competition and cannot properly estimate the important α and β parameters before the competition. During the mid-1970s, a competitive theory based on competition for resources was developed stimulated by dissatisfaction with the classical theory, the so-called resource competition model. Based on the Monod model, this model mainly focuses on the dynamical behavior while multiple populations compete for multiple resources. Tilman et al. established different consumer–resource models in [1, 2]; from then on, a large number of articles emerged, especially during the recent two or three decades (see [3–6]). Based on Tillman’s theory, scholars also proposed a new method that predicted the final competition results by using resource requirement among competing populations. However, owing to the complexity of competition among populations, that theory is still not perfect. Nowadays, the minimum requirements competition theory of Tillman’s is still popular. The theory considers that the final winner will be that population which has the minimum resource requirements. While the relative growth rate of a population is the minimum function of resources, it enhances the difficulty of research. Many researchers focused on the competition between two populations and one resource. Hsu [7] considered the disturbances from the opponents competing for resources and pointed out that the final winner among the predators depends on its initial population size. In 1999, Huisman (see [8]) went on studying the model established by Tillman in 1977. He pointed out that it was competition for resources that led to the bio-diversity, thereby studying the resources’ competition model was obvious and essential. Smith et al. (see [9]) proved, by using matrix theory, that there is no equilibrium point provided the population size exceeds the number of resources, while also considering that the relative mortality was equal to the transform rate among resources. There are many other works about this problem (see [10–15]).
Seeing the complication of the stochastic model, only the white noise is considered. This paper consists of several parts: the existence of solution is studied in Sect. 2, the stochastic final boundedness is discussed in Sect. 3, path estimation is studied in Sect. 4, the persistence and extinction are finally discussed in Sect. 5.
2 Existence of positive solutions
Theorem 1
For any given initial condition \((N_{i}(0),R_{j}(0)) \in R_{+}^{n}\times R_{+}^{k}\) \((i=1,2,\dots,n; j=1,2,\dots,k)\), there exists a unique solution \((N(t),R(t))\) (where \((N(t)=(N_{1}(t), N_{2}(t),\dots, N_{n}(t)), R(t)=(R_{1}(t), R_{2}(t),\dots,R_{k}(t)))\) for system (5)–(6), and this solution remains in \(R_{+}^{n}\times R_{+}^{k}\) with probability 1.
Proof
3 Stochastic final boundedness of the system solutions
Definition 1
Remark 1
For any resource, \(R_{j}(t)\leq S_{j}\ (1\leq j\leq k)\), so it is reasonable to consider the boundedness of resource \(R(t)\).
Lemma 1
Proof
Proof
4 Path estimation of the system solutions
Theorem 3
Proof
5 Permanence and extinction
In this part, we will discuss the situation when the solution of system (5)–(6) will be permanent or extinct under some certain conditions. For the definitions see [27].
Theorem 4
Proof
Remark 2
From (26) we know that population will become extinct when the input of resources tends to zero.
Theorem 5
Proof
6 Numerical examples
In this section we demonstrate the efficiency of the proposed condition of permanence and extinction with some illustrative examples.
Example 1
Let \(\alpha_{11}=0.12\), \(\alpha_{12}=0.145\), \(\alpha_{21}=0.15\), \(\alpha _{22}=0.1\), \(r_{1}=0.35\), \(r_{2}=0.35\), \(K_{11}=0.4\), \(K_{12}=0.5\), \(K_{21}=0.3\), \(K_{22}=0.5\), \(m_{1}=0.5\), \(m_{2}=0.5\), \(S_{1}=411.5\), \(S_{2}=411.35\), \(D=0.2\), \(C_{11}=0.15\), \(C_{12}=0.15\), \(C_{21}=0.13\), \(C_{22}=0.15\).
Let \(\alpha_{11}=0.12\), \(\alpha_{12}=0.145\), \(\alpha_{21}=0.15\), \(\alpha _{22}=0.1\), \(r_{1}=0.35\), \(r_{2}=0.35\), \(K_{11}=0.4\), \(K_{12}=0.5\), \(K_{21}=0.3\), \(K_{22}=0.5\), \(m_{1}=0.325\), \(m_{2}=0.315\), \(S_{1}=411.5\), \(S_{2}=411.35\), \(C_{11}=0.15\), \(C_{12}=0.15\), \(C_{21}=0.13\), \(C_{22}=0.15\).
Let \(\alpha_{11}=0.12\), \(\alpha_{12}=0.145\), \(\alpha_{21}=0.15\), \(\alpha _{22}=0.1\), \(r_{1}=0.35\), \(r_{2}=0.35\), \(K_{11}=0.4\), \(K_{12}=0.5\), \(K_{21}=0.3\), \(K_{22}=0.5\), \(m_{1}=0.5\), \(m_{2}=0.315\), \(S_{1}=411.5\), \(S_{2}=411.35\), \(C_{11}=0.15\), \(C_{12}=0.15\), \(C_{21}=0.13\), \(C_{22}=0.15\).
More precisely, it can be observed that the populations get extinct or will both be permanent depending on the relationship between the intensity of environmental noises \(\alpha_{i}\), death rate \(m_{i}\), transformation rate of system D and supply of resources \(S_{i}\). That is, having enough resources and a lower death rate is beneficial to the survival of the population (see Fig. 4), and on the contrary, if there is a high-intensity environmental fluctuation, the population may suffer extinction (see Figs. 1–3). Thus, the environmental noise may affect the evolution trend of a population.
7 Conclusion
Declarations
Acknowledgements
The authors would like to thank the anonymous reviewers and the editor for their valuable comments and suggestions that helped improve the manuscript.
Funding
The present investigation was supported in part by the Natural Science Foundation of China (Grant No. 11661064) and the Scientific Research Foundation of the Ningxia Higher Education Institutions of China (Grant No. NGY2017033).
Authors’ contributions
All authors made equal contributions. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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