# Permanence and extinction of a high-dimensional stochastic resource competition model with noise

## 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, 4, 5, 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, 11, 12, 13, 14, 15]).

*R*is the density of nutrients in the system,

*D*is the dilution rate or the input rate of nutrients,

*S*denotes the supply of nutrients or resources, \(\mu_{m}\) denotes the maximum birth rate,

*N*denotes the population density or size,

*K*denotes the half-saturation constant, i.e., the amount of nutrients while the birth rate is half of \(\mu_{m}\),

*Y*is the size of the produced population for individual units. There have been many results on this kind for chemostat models (see [16, 17, 18, 19]). In order to better match up the reality, Tillman generalized the original model to have

*n*populations and

*k*nutrients (resources). The specific model is as follows [20]:

*i*, \(r_{i}\) denotes the maximal growth rate, \(m_{i}\) denotes the relative death rate of population

*i*, \(N_{i}m_{i}\) denotes the death rate of population

*i*, \(R_{j}\) denotes the amount of the

*j*th available resource,

*D*denotes the transformation rate of the system, \(S_{j}\) denotes the support of the

*j*th resource, \(C_{ji}\) denotes the

*j*th resource that gets consumed by the

*i*th population, \(\sum_{i=1}^{n}C_{ji}\mu_{i}(R_{1},R_{2},\dots,R_{k})N_{i}\) denotes the total consumption by all the populations, \(DR_{j}\) denotes the self-consumption rate of the

*j*th resource. The well-known Monod function in ecology describes the relative birth rate of the population which is the function of resources; \(K_{ji}\) in the Monod function denotes the corresponding resources when the population birth rate becomes half of \(r_{i}\). Equation (2) illustrates that the birth rate of a certain population depends on the resource which has the minimum support. Equation (3) shows that the amount of the

*j*th resource depends on the support and consumption of resources.

*n*populations competing for

*k*resources. In fact, a biological system is inevitably affected by environmental noises (see [22]). May has pointed out that parameters in systems exhibited random fluctuations to a greater or lesser extent due to environmental noises [23]. Thus, it is meaningful to take environmental noises into consideration. The most important parameters for a population ecosystem are the intrinsic growth rate (= birth rate \(\mu_{i}-\mbox{ death rate }m_{i}\)), so we used the technique of parameter perturbation to examine the effect of environmental noise on intrinsic growth rate: \(\gamma_{i}=\mu_{i}-m_{i} \Longrightarrow(\mu_{i}+\alpha _{i1}\,dB_{1}(t))-(m_{i}+\alpha_{i2}\,dB_{2}(t))\). That is, the birth and death rates are subjected to a normal distribution with means \(\mu_{i}\) and \(m_{i}\). Owing to the complication of the system, we are only concerned with white noise. For many new conclusions on this kind of competition model with regime switching or impulsive effect, the readers are referred to [24, 25, 26]. In this paper, we will focus on the following model:

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

*m*. Let \(\tau_{\infty}=\lim_{m\rightarrow\infty}\tau_{m}\), and we define a twice differentiable function \(V:R_{+}^{n}\rightarrow R_{+}\) as follows:

## 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

*Let*\(\theta\in(0,1)\), \(\bar{D}=\min_{1\leq i\leq n}\{ D, m_{i}\}\), \(\gamma=\bar{D}+\frac{1}{2}(1-\theta)\max_{1\leq i\leq n}\{ \alpha_{i2}^{2}\}>0\).

*Then for any*\(\xi\in(0, \gamma\theta)\),

*there exists*\(\hat{H}>0\),

*such that the solution of system*(5)

*–*(6)

*satisfies*

### Proof

*Ĥ*such that

*t*and taking expectation, we obtain

### Proof

## 4 Path estimation of the system solutions

### Theorem 3

### Proof

*η*is a positive real number to be determined. By using Itô formula, we obtain

*J*such that

*t*the above inequality, and then taking expectation, we obtain

## 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

*Suppose that the noise intensity satisfies*\(\max_{1\leq i\leq k}\{\alpha_{i1}^{2}+\alpha_{i2}^{2}\}<2D\).

*For an arbitrary initial condition*\((N_{i}(0),R_{j}(0))\in R_{+}^{n}\times R_{+}^{k}\),

*if*\(c\max_{1\leq j \leq k}\{S_{j}\}-m_{i}-\frac{1}{2}(\alpha_{i1}^{2}+\alpha _{i2}^{2})<0\)\((1\leq i \leq n)\),

*then the solution of system*(5)

*–*(6)

*satisfies*

*that is*,

*the solution of system*(5)

*–*(6)

*will exponentially fast become extinct almost surely*.

*Here*

*c*

*is a positive constant*,

*satisfying*\(\max_{{1\leq i \leq n}, {1\leq j \leq k}}\frac{r_{i}}{K_{ji}+R_{j}}= c\).

### Proof

*t*both sides of (11) and then dividing by

*t*, we get

*t*both sides of the above formula and then dividing by

*t*, we get

### Remark 2

From (26) we know that population will become extinct when the input of resources tends to zero.

### Theorem 5

*Assume that the noise intensity satisfies*\(\max_{1\leq j\leq k}\{\alpha_{i1}^{2}+\alpha_{i2}^{2}\}<2D\).

*Then for any given initial condition*\((N_{i}(0),R_{j}(0))\in R_{+}^{n}\times R_{+}^{k}\),

*if*\(c_{i}\min_{1\leq j \leq k}\{S_{j}\}-m_{i}-\frac{1}{2}(\alpha_{i1}^{2}+\alpha _{i2}^{2})>0\)\((1\leq i \leq n)\),

*system*(5)

*–*(6)

*satisfies*

*that is*,

*the solution of system*(5)

*–*(6)

*will be persistent in the mean*.

### 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

*n*populations competing for

*k*necessary resources. By using stochastic analysis, stochastic final boundedness of the

*i*th population, moment boundedness and extinction or permanence under certain conditions in the system (5)–(6) are obtained. It is found that the requirement of white noise is identical with those in existing results, that is, populations will get extinct when the noise is very strong. Furthermore, a path estimate of the

*i*th population is also obtained. For resources’ competition system, the birth rate of the population described by the minimum function is indeed affected by the number of resources, which is compatible with the known theory, in which those who have the least resource consumption will maintain persistence. However, as we know, there are many different random perturbations that should be considered, such as the telephone noise, Levy noise, etc. Due to the complexity of the system with

*n*populations competing for

*k*resources, in this paper, we only consider the white noise, however, we can consider the Markovian switching into model (5)–(6) in future, which takes the following form:

## Notes

### Acknowledgements

The authors would like to thank the anonymous reviewers and the editor for their valuable comments and suggestions that helped improve the manuscript.

### Authors’ contributions

All authors made equal contributions. All authors read and approved the final 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).

### Competing interests

The authors declare that they have no competing interests.

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