Randomly switching evolution equations☆
Introduction
The theory of piecewise deterministic Markov processes (PDMP’s) has generated considerable interest in the scientific community over the past three decades, having been first introduced in [1]. From the point of view of modelling in natural sciences the class of PDMP’s is a very broad family of stochastic models covering most of the applications, omitting mainly diffusion related phenomena. A recent monograph [2] surveys the applicability of PDMP’s to problems in the biological sciences.
Briefly, a PDMP is a continuous-time Markov process with values in some metric space. The process evolves deterministically between the so-called jump times that form an increasing sequence of random times. Usually deterministic evolution is described by ordinary differential equations (ODE’s) inducing dynamical systems (or flows). However, in a PDMP at the jump times one considers a different type of behaviour such that there is an actual jump to a different point in the phase space or a change of the dynamics. The latter are referred to as randomly switching dynamical systems or switching ODE’s.
The class of PDMP’s is widely used in many areas of science, especially in biology [2], [3], and these include the applications of randomly switching dynamical systems. A prototype of PDMP’s is the telegraph process first studied by Goldstein [4] and Kac [5] in connection with the telegraph equation where a particle moves on a real line with constant velocity alternating between two opposite values according to a Poisson process. An extension of such a process is a velocity jump process where an individual moves in a space with constant velocity and at the jump times a new velocity is chosen randomly [6], [7]. Another example is a multi-state gene network where the gene switches between its active and inactive state at the jump times [8], [9], [10].
Existing models usually describe the underlying phenomena for some population from the point of view of a single individual. In physics this is often known as a particle perspective [11]. That means that the dynamics of every single individual is driven by separate stochastic laws depending on a variety of factors, e.g. its mass or energy. However, there are alternative situations in which the entire population is affected by randomly switching environmental conditions, e.g. particles driven by a common environmental noise [11] or the response of a metabolic or gene regulatory system to an environmental stimulus [12]. This is known as a population perspective [11], and is the approach that we use in this paper since we consider one common source of randomness which affects all individuals in a population.
Here we treat the evolution of the density of a population distribution in the situation where every individual has its own deterministic dynamics but the whole population is affected by some continuous-time Markov process with finite state space that changes the current state of all individuals. In this approach, a state is represented by a population density — an element of an infinite dimensional space. It is particularly difficult to study the evolution of such densities and thus we investigate their moments and correlations of all orders. These infinite dimensional processes are dual to the class of PDMP’s known as random evolutions introduced earlier by Griego and Hersh [13], motivated by the work of Goldstein [4] and Kac [5], see [14]. (For an amusing and non-technical account of this history see [15] and [16].) They are particular examples of models governed by so-called switching Partial Differential Equations (PDEs) that recently have a growing interest in the literature [17], [18], [19], [20], [21]. Most of these papers focus on applications in biological sciences. From the mathematical point of view they are based on diffusion processes and PDEs of parabolic type.
In [17], [22] the authors provide the moment and correlation equations in the case of diffusion processes. The current study is a generalization of their work by giving moment and correlation equations for a broader class of processes. The main result of our paper is that the mean of a process described by randomly switching PDEs can be viewed as an appropriate stochastic semigroup (see Theorem 5.1 and Corollary 5.3). This has further important consequences, especially that the mean of random density in the population perspective can be seen as identical to a density from the individual perspective (see Section 6). We study the mean of the process at large time for a variety of examples that are biological applications. It allows us to investigate the asymptotic behaviour for the mean of the process in the cases of fold, transcritical, pitchfork, and Hopf bifurcations. We also provide numerical simulations for the mean of the process which were prepared by using FiPy [23].
This paper is organized as follows. In Section 2 we provide some basic material from the theory of stochastic semigroups on . Section 3 briefly reviews randomly switching dynamical systems in Euclidean state spaces. In Section 4 we introduce randomly switching semigroups with the state space being the set of densities leading to a stochastic evolution equation in an space. We study the first moment of its solutions in Section 5 where we stress the correspondence between this moment equation and the Fokker–Planck type equations from Section 4. Section 5 contains the main results of this paper, namely Theorem 5.1 and Corollary 5.3. The behaviour of the mean at large time is considered in Section 6, where we also give examples of applications of our results to situations in which the underlying dynamics display a variety of bifurcations. In Section 7 we study second order correlations of solutions of the stochastic evolution equation. We conclude in Section 8 with a brief summary. The appendix contains relevant concepts from the theory of tensor products that are used in Section 7.
Section snippets
Preliminaries
In this section we collect some preliminary material. We begin with the notion of stochastic (Markov) semigroups and provide examples of such semigroups.
Let a triple be a -finite measure space and let . We define the set of densities by A stochastic (Markov) operator is any linear mapping such that [24]. A family of linear operators on is called a stochastic semigroup if each operator is stochastic and is a
Randomly switching dynamics
In this section we recall a classical setting of PDMP based models seen from the perspective of individual units and taking place in a finite dimensional space. This well-known situation will be contrasted in the following sections with a population perspective approach, thus moving the analysis to infinite dimensional space. See [11, Figure 1] for a nice pictorial distinction between the individual and populational perspectives.
Consider sufficiently smooth vector fields , , ,
Randomly switching densities
In this section we look at the role of stochasticity in explaining biological phenomena from the point of view of the whole population in an environment affected by some random disturbances. We illustrate this approach using two examples, namely a population model with two different birth rates [26], and a model of inducible gene expression with positive feedback on gene transcription [27].
Example 4.1 Population Model with Two Different Birth Rates Consider a population of size , a death rate and birth rate with changing in time between two
First moment equations
We continue with the general setting from Section 4 and study the first moment of the solution of (4.5). Recall that , where is given by (4.8). Using a simple decomposition, the first moment of can be written as where denotes the expectation and is the indicator function of an event . If we take then We will show that:
Mean of the process at large time
We consider the relationship between Fokker–Planck type systems (3.5) for a distribution of processes in space, and the first moment equation (5.3). The latter has the same form as (3.6) and is the solution of the evolution equation (3.5) with initial condition . Hence, if is a solution of (3.6) and for each there exists such that then, by Corollary 5.3, we have and, by Eq. (5.2),
Second and higher order correlations
In this section we continue the study of the stochastic process (4.6) by looking at equations for correlations. These are extensions of the moment equations considered in Section 5. We provide the full analysis only for second order correlations, but higher order cases are straightforward and can be easily obtained by similar considerations. We use some notation from the theory of tensor products, and for a brief summary of standard definitions used here see Appendix.
We start with the
Conclusion
In this paper we introduced the concept of randomly switching stochastic semigroups. We investigated a stochastic evolution equation in space. Such a regime could explain the source of stochasticity when observing the evolution of some population driven by a common environmental stimulus. Next, we studied the first moment of the stochastic evolution equation solutions and found the correspondence between this moment equation and a deterministic system of Fokker–Planck type equations for the
CRediT authorship contribution statement
Paweł Klimasara: Conceptualization, Methodology, Writing and editing of the manuscript. Michael C. Mackey: Conceptualization, Methodology, Writing and editing of the manuscript. Andrzej Tomski: Conceptualization, Methodology, Writing and editing of the manuscript. Marta Tyran-Kamińska: Conceptualization, Methodology, Writing and editing of the manuscript.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We would like to thank Ryszard Rudnicki and Radosław Wieczorek for helpful discussions. The authors are especially appreciative of the comments of three anonymous referees that materially improved the presentation. This research was supported in part by the Natural Sciences and Research Council of Canada (NSERC) and the Polish NCN grant 2017/27/B/ST1/00100.
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This research was supported in part by the Natural Sciences and Research Council of Canada (NSERC) and the Polish NCN grant 2017/27/B/ST1/00100.