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A time-delayed tumor cell growth model with correlated noises is investigated. In the condition of small delay time, the stationary probability distribution is derived and the stationary mean value (〈x〉st) and normalized varianceλ2 of the tumor cell population and state transition rate (κ) between two steady states are numerically calculated. The results indicate that: (i) The delay time (τ) enhances the coherence resonance in 〈x〉st as a function of the multiplicative noise intensity (D) and increases 〈x〉st as a function of the additive noise intensity (α), i.e., τ enhances fluctuation of the system, however, the strength (λ) of correlations between multiplicative and additive noise plays a contrary role to τ on these; (ii) τ enhances the coherence resonance in κ as a function of D and increases κ as a function of α, i.e., τ speeds up the rate of state transition, however, λ also plays a contrary role to τ on these.

Logistic growth is one of the most popular equations not only in mathematical ecology but also in other applications. First introduced by Verhulst for saturated proliferation of a single-species [1], it has been extended to include spatial dynamics by Fisher [2] and by Kolmogoroff et al. [3]. It is now one of the classical examples of self-organization in many natural and artificial systems [4]. In this paper we focus on time delay effects on the basis of the logistic growth equation. In particular, this equation was proposed to describe the growth of the Ehrlich ascities tumor (EAT) in a mouse [5]. It appears that even such a simple ordinary differential equation can be used to model such a complicated process. The equation of the logistic growth model reads equation(1) View the MathML sourcedxdt=ax−bx2. Turn MathJax on Here xx is the tumor mass (or denotes tumor cell population), so is confined to positive real numbers, aa is the cell growth rate and bb is the cell decay rate. Previously, much attention has been paid to the statistical properties of the tumor cell growth model with correlated noises [6], [7], [8], [9], [10] and [11]. It was found that the correlations between noises essentially affect the stationary and transient properties of the model. However, in previous works, the members of the population were assumed to react instantaneously to any change in the environment; a time delay was not included in the logistic growth model. For population dynamics, there should be a reaction time of the population to environmental constraints in the process of the population evolution [12] and [13]. Time delay can play a crucial role in the modeling of biological processes [14], especially on the cellular level, which is extremely important in tumor growth modelling [15] and [16]. Thus, the logistic growth model with a time delay driven by cross-correlation noises should be investigated. In this paper, we investigate the effects of time delay in a logistic growth model with the combination of correlated noises with nonzero correlation time and time delay. In Section 2, we derive the approximate Fokker–Planck equation and the stationary probability distribution (View the MathML sourceSPD) of the model, and calculate the mean (View the MathML source〈x〉st) of the population and the state transition rate (κκ) between two stable states of the system. Section 3 consists of a discussion and the conclusion to the paper.

Based on considering the reaction time of the population to environmental constraints, we introduce a time delay into the logistic growth model with correlated noises. We derived an approximate Fokker–Planck equation and the SPDFSPDF of the system. Making use of the SPDFSPDF, we investigated the effects of ττ and λλ on the stationary properties and state transition rate of the tumor cell growth model. The delay time ττ enhances the coherence resonance in View the MathML source〈x〉st as a function of the multiplicative noise intensity DD, and increases the View the MathML source〈x〉st as a function of the additive noise intensity αα, i.e., ττ enhances fluctuation of the system, however, the strength λλ of correlations between multiplicative and additive noise plays a contrary role to ττ on these factors. ττ enhances the coherence resonance in κκ as a function of DD, and increases the κκ as a function of αα, i.e., ττ speeds up the rate of state transition, however, λλ also plays a contrary role to ττ on these factors.