Due to climate change and an increase of favourable habitat, ticks and tick-borne diseases are reported to expand to northern areas in north America. One main factor for lone star ticks to be established in Canada is due to the migration of white-tailed deers from US. In this work, we formulate a compartmental model to study the dynamics of lone star ticks and white-tailed deers, with a focus on migration effect of white-tailed deers. The tick-host interaction and the effect of deer migration are explored analytically and numerically. The positivity of the populations in the model is proved, and the unique positive equilibrium is proved to be asymptotically stable. We conduct sensitivity analysis on a set of parameters, revealing the correlation between the parameters and equilibrium populations. Numerical results show that migration rate of white-tailed deer is one crucial parameter that increases the populations of (infected) ticks and (infected) hosts.

In this paper, the formats of Julia sets for a class of nonlinear complex dynamic systems with variable coefficients were studied under certain conditions. For the complex dynamic systems in piecewise cases, we proposed some methods to control the forms of their Julia sets and stable domains analytically. What's more, we illustrated that our methods worked well by computational simulations. Our work provides a better understanding about how to control the Julia sets of certain complex dynamic systems.

It is widely-held belief that people's consumption behavior is partly determined by advertising and word-of-mouth effect especially in monopolistic competitive market. Owing to this, we propose a mathematical model to interpret consumer behavior under the advertising and word-of-mouth effects, which is divided into continuous and discrete types for dynamic behavior analysis. Our research indicates the continuous model undergoes fold bifurcation, Hopf bifurcation, and degenerate fold-Hopf bifurcation; the discrete model undergoes flip bifurcation and Neimark-Sacker bifurcation. Moreover, bifurcation diagrams are given by using MATLAB to illustrate the model. Based on the theory of Hopf bifurcation or Flip bifurcation, the system undergoes supercritical Hopf bifurcation or Flip bifurcation under certain conditions. This leads to both advertising impacts periodically on consumer behavior, and a short-term reduction cause no effect the public's recognition of the brand, which can guide precision advertising investment.

It is well-known that the transmission of malaria is caused by the bites of mosquitoes. Since the life habit of mosquitoes is influenced by seasonal factors such as temperature, humidity and rainfall, the transmission of malaria presents clear seasonable changes. In this paper, in order to take into account the incubation periods in humans and mosquitoes, we study the threshold dynamics of two periodic reaction-diffusion malaria models with distributed delay in terms of the basic reproduction number. Firstly, the basic reproduction number $R_0$ is introduced by virtue of the next generation operator method and the Poincaré mapping of a linear system. Secondly, the threshold dynamics is established in terms of $R_0$. It is proved that if $R_0 < 1$, then the disease-free periodic solution of the model is globally asymptotically stable; and if $R_0 > 1$, then the disease is persistent.

Based on T. Tao's celebrated result on the norm convergence of multiple ergodic averages for commuting transformations, we find that there is a subsequence which converges almost everywhere. Meanwhile, we obtain the ergodic behaviour of diagonal measures, which indicates the time average equals the space average. According to the classification of transformations, we also give several different results. Additionally, on the torus $\mathbb{T}^d$ with special rotation, we prove the pointwise convergence in $\mathbb{T}^d$ , and get a result for ergodic behaviour.

In this paper, we investigate qualitative properties of crossing limit cycles for a class of discontinuous nonlinear Liénard-type differential systems with two zones separated by a straight line. Firstly, by applying left and right Poincaré mappings we provide two criteria on the existence, uniqueness and stability of a crossing limit cycle. Secondly, by geometric analysis we estimate the position of the unique limit cycle. Several lemmas are given to obtain an explicit upper bound for the amplitude of the limit cycle. Finally, a predator-prey model with nonmonotonic functional response is studied, and Matlab simulations are presented to show the agreement between theoretical results and numerical analysis.

In this paper, we deal with a diffusive predator-prey model with mixed boundary conditions, in which the prey population can escape from the boundary of the domain while predator population can only live in this area and can not leave. We first investigate the asymptotic behaviour of positive solutions and obtain a necessary condition ensuring the existence of positive steady state solutions. Next, we investigate the existence of positive steady state solutions by using maximum principle, the fixed point index theory, $L_p$-estimation, and embedding theorems, Finally, local stability and uniqueness are obtained by linear stability theory and perturbation theory of linear operators.

This paper study the planar quadratic semi-quasi-homogeneous polynomial systems (short for PQSQHPS). By using the nilpotent singular points theorem, blow-up technique, Poincaré index formula, and Poincaré compaction method, the global phase portraits of such systems in canonical forms are discussed. Furthermore, we show that all the global phase portraits of PQSQHPS can be classed into six topological equivalence classes.

This paper investigates the effects of periodic drug treatment on a HIV infection model with two co-circulation populations of target cells. We first introduce the basic reproduction ratio for the model, and then show that the infection free equilibrium is globally asymptotically stable if $\mathcal{R}_0 < 1$, while the infection persists and there exists at least one positive periodic state when $\mathcal{R}_0 > 1$. Therefore, $\mathcal{R}_0$ serves as a threshold parameter for the infection. We then consider an optimization problem by shifting the phase of drug efficacy functions, which corresponds to change the dosage time of drugs in each time interval. It turns out that shifting the phase affects critically on the stability of the infection free steady state. Finally, exhaustive numerical simulations are carried out to support our theoretical analysis and explore the optimal phase shift.

The multiplicity and stability of the equilibrium states of a three-dimensional differential system with initial conditions and three cross terms are studied in this paper. The existence and multiplicity of equilibrium states are given under the different qualifications of parameters. Besides, the local stability of the equilibrium state is shown by analyzing the eigenfunction of the Jacobi matrix. The global stability of the equilibrium state is obtained by constructing the Lyapunov function. Furthermore, the numerical simulation intuitively reflected the relationship of variables and verified the correctness of theoretical analysis.