Whale optimization algorithm (WOA) is a relatively new population-based metaheuristic optimization method, which has the advantage of fewer control parameters, strong global optimization ability and easy to implement. However, when being used for high-dimensional problems, WOA may be trapped in the local optimum. In this study, we propose an effective whale optimization algorithm called EWOA. Inspired by particle swarm optimization (PSO), a modified position-updated equation by introducing dynamic inertia weight parameter to guide the search of new candidate individuals is presented. In addition, in order to make full use of and balance the exploration and the exploitation of WOA, a nonlinea... More >

This article examines the dynamic behavior of a second-order fuzzy difference equation that models the quantitative changes in a specific biological population: $$ E_{n+1}=\frac{S}{C+E_n+E_{n-1}},\ n\in \mathbb{Z}\ and\ n\ge 0,$$ Here, parameter $S$ represents the carrying capacity of the environment, while $C$ signifies the minimum resources required for population survival. The initial values $E_0$, $E_{-1}$, and parameters $S$ , $C$ are all positive fuzzy numbers. By employing the generalized division (g-division) with respect to fuzzy numbers, we establish the existence, uniqueness, persistence, and boundedness of positive fuzzy solutions to the equation under specified conditions. Furth... More >

A class of multi-age group $SEQ_3IHR$ chickenpox epidemic models with effective vaccination in a different subpopulation size, is considered, in this paper. It is assumed that individual members of the subpopulations who recovered from the disease, may be re-infected by the disease. The models comprise of six infectious classes of subclasses and three distinct infected stages of subclasses. We establish that the global dynamics of the epidemic models are determined entirely by the basic reproduction number. In this paper, we determine that the disease-free and endemic equilibrium states, and carry out a stability analysis of the disease-free as well as the endemic equilibrium points of the m... More >

In the evolutionary game, the introduction of the third or more specific strategies to study cooperative evolution has attracted widespread attention recently. The main feature of this paper is to adopt a special reward strategy as the third choice to observe the evolution dynamics. This strategy can improve the utilization of resources and help cooperators and defectors get greater benefits. We investigate the evolutionary game dynamics of a three-strategy snowdrift game with special reward strategy and payoff delay. First, for the nondelay system, we discuss the dynamic properties of strategies, including the existence, stability and bistability of equilibrium states. Furthermore for the t... More >

This article develops a constructive approach to the Painlevé equivalence problem based on explicit normalization and residual obstruction analysis. Instead of starting from the full invariant complexity of the general point-equivalence problem, we use target-adapted reductions that convert equivalence into a finite algebraic comparison problem inside canonical normalized families. For derivative-free polynomial classes, the method yields exact results. We obtain a constructive criterion for equivalence to the first Painlevé equation in the quadratic case and an exact criterion for equivalence to the second Painlevé equation in the cubic case. In both settings, the method provides explici... More >

In this paper, we present an important property of smooth function germs of orbit tangent space, and apply it to discuss the proof of Morse's Lemma. More >

In prostate cancer treatment, intermittent androgen deprivation therapy can delay the emergence of drug resistance, yet its efficacy is influenced by environmental fluctuations, treatment parameters, and competition among tumor cell populations. To investigate the mechanisms underlying resistance during intermittent therapy, we developed a stochastic impulsive differential equation model. First, a global analysis of the model without environmental noise is conducted. Subsequently, sufficient conditions for tumor extinction and persistence are established based on stochastic stability theory. Numerical simulations further demonstrate that moderate noise suppresses tumor growth, and an optimal... More >

This article delves into the trajectory structure rules of a specific fifth-order rational difference equation: $$ s_{m+1}=\frac{s_ms_{m-2}s_{m-3}s_{m-4}+s_ms_{m-2}+s_ms_{m-3}+s_{m-2}s_{m-3}+s_{m-4}+a}{s_ms_{m-2}s_{m-3}+s_ms_{m-2}s_{m-4}+s_ms_{m-3}s_{m-4}+s_{m-2}s_{m-3}s_{m-4}+1+a} $$ where the initial conditions satisfy $s_i\in (0,\infty)$, $i=-4,-3,-2,-1,0$, and the parameters $a\in [0,\infty).$ As the initial values vary, the lengths of consecutive positive and negative semi-cycles for non-trivial solutions exhibit a periodic pattern with a prime period of 31. The rule within one period is $1^-, 2^+, 1^-, 1^+, 1^-, 1^+, 2^-, 4^+, 3^-, 2^+, 2^-, 1^+, 5^-, 1^+, 1^-,$ $ 3^+ $. Through t... More >