New insights in stability analysis of delayed Lotka–Volterra systems

Abstract

Most of the reported Lotka–Volterra examples have at most one stability interval for the delay parameters. Furthermore, the existing methods fall short in treating more general case studies. Inspired by some recent results for analyzing the stability of time-delay systems, this paper focuses on a deeper characterization of the stability of Lotka–Volterra systems w.r.t. the delay parameters. More precisely, we will introduce the recently-proposed frequency-sweeping approach to study the complete stability problem for a broad class of linearized Lotka–Volterra systems. As a result, the whole stability delay-set can be analytically determined. Moreover, as a significant byproduct of the proposed approach, some Lotka–Volterra examples are found to have multiple stability delay-intervals. To the best of the authors’ knowledge, such a characterization represents a novelty for having some insights in the population dynamics: in some situations, a longer maturation period of species is helpful for the stability of a population system.

Introduction

Since the pioneering work by Lotka [1] and Volterra [2] in the 1920s, the Lotka–Volterra models have been extensively used to describe population dynamics. It is now recognized that incorporating some delay factors into a classical Lotka–Volterra model is more realistic. For instance, in describing the predator–prey dynamics, the gestation period of species, the hunting time lag of predator to prey, and maturation time of species should be appropriately considered. All these factors introduce delays in the Lotka–Volterra models.

The Lotka–Volterra systems with delays, or called delayed Lotka–Volterra systems, have attracted the interest of a wide range of researches (in applied mathematics, cybernetics, biology, ecology, etc.) for the past decades. In particular, the stability of delayed Lotka–Volterra systems has been an active research topic, since May [3] first proposed and discussed the stability of a delayed Lotka–Volterra model of two species in 1973.

Following a standard stability-analysis procedure, the local stability analysis for a Lotka–Volterra system can be recast into studying the distribution of the characteristic roots. In particular, for a large number of Lotka–Volterra systems encountered in the literature, the corresponding characteristic functions are in the quasipolynomial form f(λ, τ), where λ is the Laplace variable and τ is the only delay parameter.

It is known that if all the characteristic roots for $f(\lambda ,\tau )=0$ are located in the left half-plane ${\mathbb{C}}_{-},$ then the associated Lotka–Volterra system is locally asymptotically stable at the positive equilibrium. Thus, for a broad class of Lotka–Volterra systems, we can analytically study the local asymptotic stability.

As already verified by various examples, a generic time-delay system may have more than one stability interval for the delay parameter. Hence, the importance of the complete stability analysis w.r.t. the delay (stability analysis for a time-delay system along the whole semi-infinite positive delay axis) is clear, especially when we use the delay as a control parameter (see e.g., [4], [5], and the references therein).

This inspires us to figure out whether a Lotka–Volterra system may have more than one stability delay-interval. In our opinion, such a task is not easy for two reasons: (1) compared with a generic time-delay system, the coefficients for a Lotka–Volterra system have to meet some additional constraints (this point will be demonstrated through two Lotka–Volterra models in this paper). (2) the mathematical tools used in the existing literature have some limitations (some details will be found in Section 5.1) and there is still some room to improve the stability studies for delayed Lotka–Volterra systems and to find new properties for the corresponding dynamic behavior.

In this paper, we will take two commonly-encountered Lotka–Volterra models for illustrations. It is worth mentioning that the approach used in this paper is not limited to specific models and it can be applied to the general Lotka–Volterra systems.

As for the tool of this paper, we will introduce the frequency-sweeping approach (recently proposed in [6], [7]), with which we are able to systematically solve the associated complete stability problem.

It is an important contribution of this paper that some Lotka–Volterra examples with multiple stability delay-intervals are found. These examples will be studied in detail and verified by simulations. Such examples depict a new interesting feature: a longer maturation period may be beneficial for the species stability. Some detailed analysis from the angle of biology/ecology will be presented in Section 4.2. Although the topics concerning population dynamics (one may refer to the monographs [8], [9], [10]) and population dynamics with delays (one may refer to the monographs [4], [11]) are relatively well investigated in the literature, to the best of the authors’ knowledge the above feature has not been reported.

This paper is organized as follows. Some preliminaries and prerequisites are given in Section 2. The frequency-sweeping approach is introduced in Section 3. Some Lotka–Volterra examples with multiple stability delay-intervals are presented and discussed in Section 4. In Section 5, some future extensions are discussed. Finally, the paper concludes in Section 6.

Notations: Throughout the paper, the following standard notations are used: $\mathbb{C}$ is the set of complex numbers. ${\mathbb{C}}_{-}$ and ${\mathbb{C}}_{+}$ denote the left half-plane and the right half-plane in $\mathbb{C},$ respectively. ${\mathbb{C}}_0$ is the imaginary axis and $\partial \mathbb{D}$ is the unit circle. $\mathbb{N}$ and ${\mathbb{N}}_{+}$ are the sets of non-negative integers and positive integers, respectively. ε is a sufficiently small positive real number. For $\gamma \in \mathbb{R},$ ⌈γ⌉ denotes the smallest integer greater than or equal to γ. Finally, $\mathrm{deg}(·)$ denotes the degree of a polynomial.