Stochastic competitive Lotka–Volterra ecosystems under partial observation: Feedback controls for permanence and extinction

Abstract

This paper is concerned with Lotka–Volterra models formulated using stochastic differential equations with regime switching represented by a continuous-time Markov chain. Different from the existing literature, the Markov chain is hidden and can only be observed in a Gaussian white noise in our work. For such partially observed problems, we use a Wonham filter to estimate the Markov chain from the observable evolution of the given process, and convert the original system to a completely observable one. We then establish the regularity, positivity, stochastic boundedness, and sample path continuity of the solution. Moreover, stochastic permanence and extinction using feedback controls are investigated. Numerical experiments are conducted to validate the theoretical findings and demonstrate how feedback controls perform in practice.

Introduction

There have been resurgent efforts in treating partially observed systems in the control and systems community. This work is devoted to a class of such systems that have been around for many years, but got more recent attentions owing to the new modeling perspective for complex systems with random environment. Assuming the random environment can only be observed with noise focusing on Lotka–Volterra systems, we develop a new approach that will be of interest not only to the researchers in ecology and bio-systems, but also for control theorists, operations researchers, and people who are working in system biology.

The traditional Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, nonlinear, differential equations, which are frequently used to describe the dynamics of biological or ecological systems in which two species interact, one as a predator and the other as prey. Initially proposed in 1910 by Lotka in the theory of autocatalytic chemical reactions [17], the equations were used to model predator–prey interactions [18] in 1925. The rationale is that when two or more species live in proximity and share the same basic requirements, they usually compete for resources, food, habitat, or territory. In reference to the study of the systems in the literature, this work develops asymptotic analysis of Lotka–Volterra models when random environment has to be taken into consideration. In particular, we treat the case that the random environment is given by a hidden Markov chain in continuous time.

Owing to the importance, the Lotka–Volterra models have received considerable attentions from multi-disciplinary communities such as biology, ecology, dynamic systems, and control and systems theory among others. There is a vast literature associated with the models. Along with the development of the deterministic models (see [7], [18], [30]), increasing attentions have placed on the stochastic counter part that enables the consideration of randomly perturbed systems. As pointed out in [23], [24] (see also [3], [4]), population models should contain a multiplicative noise term, taking into account of the interaction of the ecosystem with the environment. The interaction between noise and nonlinear determinism in ecological dynamics adds an extra level of complexity and can give rise to the complex behavior of the system, which becomes very sensitive to initial conditions, various deterministic external perturbations, and to fluctuations always present in nature (see [26], [29]).

Because of the recent effort in modeling systems with both continuous dynamics and discrete events, the so-called hybrid models have gained much popularity. A trend of effort is to depict the random environment that cannot be described by stochastic differential equations using random switching processes; see for example, [14], [19], [35], [36] among others; see also [21], [34] for a comprehensive treatment of switching processes.

The main issues concerning such systems include: Under what conditions, do the systems have global solutions? When will the systems be stable? Whether the systems are stochastically bounded? Whether or not the systems are stochastically permanent? Under what conditions, the species will extinct? If there is a tendency of extinction, can we find feedback controls so that this extinction be suppressed. More specifically, for $i=1,2,\dots ,n$, let x_i(t) be the population size of the ith species in the ecosystem at time t, denote $x\left(t\right)=\left(x_1\right(t),\dots ,x_n(t\left)\right)\prime \in {\mathbb{R}}^n$ (where $z\prime$ denotes the transpose of z for $z\in {\mathbb{R}}^{l_1\times l_2}$ with $l_1,l_2\ge 1$). Consider a competitive Lotka–Volterra model in random environments with n species given by

$$\mathit{dx}\left(t\right)=\mathrm{diag}\left(x_1\right(t),\dots ,x_n(t\left)\right)\left\{\right[b\left(\alpha \right(t\left)\right)-A\left(\alpha \right(t\left)\right)x\left(t\right)]\mathit{dt}+\Xi (\alpha )○\mathit{dw}(t\left)\right\},$$

and a constant initial condition $x\left(0\right)=x_0$. In the model, $w(·)=\left(w_1\right(·),\dots ,w_n(·\left)\right)\prime$ is an n-dimensional standard Brownian motion, and $b\left(\alpha \right)=\left(b_1\right(\alpha ),\dots ,b_n(\alpha \left)\right)\prime$, $A\left(\alpha \right)=\left(a_{\mathit{ij}}\right(\alpha \left)\right)$, $\Xi \left(\alpha \right)=\mathrm{diag}\left({\sigma }_1\right(\alpha ),\dots ,{\sigma }_n(\alpha \left)\right)$, $\alpha \in \mathcal{M}=\{1,\dots ,m\}$ represent the different intrinsic growth rates, the community matrices, and noise intensities in different external environments, respectively; $\alpha \left(t\right)$ is a finite state Markov chain. The above formulation is seen to be in the sense of Stratonovich integral. This form is often considered to be more suitable for environmental modeling (see [8]). For the stochastic differential equations in Stratonovich form, we refer to [27]; see also [9], [25] for explanations why Stratonovich integral are more suitable for modeling in many applications.

Denote

$$r_i\left(\alpha \right)=b_i\left(\alpha \right)+\frac{1}{2}{\sigma }_i^2\left(\alpha \right),r\left(\alpha \right)=\left(r_1\right(\alpha ),\dots ,r_n(\alpha \left)\right)\prime \in {\mathbb{R}}^n,\mathrm{diag}\left(x\right)≔\mathrm{diag}(x_1,\dots ,x_n),x=(x_1,\dots ,x_n)\prime \in {\mathbb{R}}^n.$$

Then the equivalent system in Itô sense is as follows:

$$\mathit{dx}\left(t\right)=\mathrm{diag}\left(x\right(t\left)\right)\left[r\right(\alpha \left(t\right))-A(\alpha \left(t\right)\left)x\right(t\left)\right]\mathit{dt}+\mathrm{diag}\left(x\right(t\left)\right)\Xi \left(\alpha \right(t\left)\right)\mathit{dw}\left(t\right).$$

The population model (1.3) was proposed and studied in detail in [35], [36]. A question naturally arises in practice: Can we design feedback controls so that the resulting system becomes permanent or extinct if we only control a partially observed system? In particular, an important problem concerns that the Markov chain $\alpha \left(t\right)$ is unobservable. That is, at any given instance, the exact state of residency of the Markov chain is not known. Thus, we cannot see $\alpha \left(t\right)$ directly but only have noise-corrupted observation in the form of $\alpha \left(t\right)$ plus noise. Such scenarios frequently arise in the real world. Taking this fact into account, in our previous work [28], we consider the case $\Xi \left(\alpha \right)=\Xi$ being independent of $\alpha$ and the population process x(t) represents the noisy observation-hidden Markov chain observed in white noise. We then used estimation schemes by means of the observable process x(t). Distinct from that work, here we suppose that the diffusion matrix $\Xi \left(\alpha \right)$ depends on environments. If we consider partially observed systems and use Wonham׳s filter similar to [1], [28], a problem arises since the filter is no longer finite dimensional. To be able to treat models in which the diffusion coefficients depend on the Markov chain, we consider Eq. (1.3) in which the Markov chain can only be observed in a Gaussian white noise. In addition, we consider the model with a control built in. Consider the controlled population system

$$\mathit{dx}\left(t\right)=\mathrm{diag}\left(\right(t\left)\right)\left[r\right(\alpha \left(t\right))-A(\alpha \left(t\right)\left)x\right(t)+u(t\left)\right]\mathit{dt}+\mathrm{diag}\left(x\right(t\left)\right)\Xi \left(\alpha \right(t\left)\right)\mathit{dw}\left(t\right),$$

and

$$\mathit{dy}\left(t\right)=f\left(\alpha \right(t\left)\right)\mathit{dt}+\beta \left(t\right)\mathit{dB}\left(t\right),y\left(0\right)=0,$$

where $\beta (·):[0,\infty )\mapsto \mathbb{R}$ is a continuously differentiable function satisfying ${\inf }_{t\ge 0}\beta \left(t\right)>0$, $f:\mathcal{M}\mapsto \mathbb{R}$ is a real-valued function, B(t) is a one-dimensional standard Brownian motion being independent of w(t), and $u\left(t\right)=\left(u_1\right(t),\dots ,u_n(t\left)\right)\prime \in {\mathbb{R}}^n$ is a feedback control.

For control problems of such partially observed systems, it is essential to converted them to completely observed ones, which can be done by using a Wonham filter. For results on the Wonham filter, we refer the reader to [31], [32]. Numerical results, including sample means and variances, assessment of approximation errors for Wonham׳s filter are presented in [33]. In the literature, the Wonham filters have been used widely to investigate control problems with partial observations; see [1], [5], [31] for applications in engineering science and finance, respectively. For related uses of hidden Markov chains and filtering theory in ecology and biology, we refer the readers to [6], [12] and references therein. Many interesting experimental results concerning competitive Lotka–Volterra can be found in [2].

In contrast to the existing results, our new contributions in this paper are as follows. (i) We use Wonham׳s filter to build a stochastic competitive Lotka–Volterra ecosystem when the Markov chain is only observable in white Gaussian noise. (ii) We convert the partially observed systems to a fully observed system by replacing the unknown Markovian states by their posterior probability estimates. (iii) We establish a number of essential biological properties of the solution including regularity and positivity, stochastic boundedness, path continuity, asymptotic properties, permanence, and extinction. (iv) We show how to design feedback controls to make a population system permanent or extinct when the Markov chain is only observed in white noise.

The rest of the work is organized as follows. Section 2 begins with the preliminaries and problem formulation, where Wonham׳s filter is introduced and the partially observed models are converted to completely observable ones. Section 3 is devoted to the suppression of population expression and biologically essential properties of the solution. Section 4 considers stochastic permanence and extinction. Feedback controls are investigated in Section 5 and numerical examples are provided in Section 6. Finally, the paper is concluded with some concluding remarks.