The cyclical dynamics of lemmings in the Arctic have garnered increasing attention over the years. The potential loss of the lemming cycle raises concerns about the collapse of the entire food chains which may include specialized predators (such as the stoat), and generalist predators (arctic fox, snowy owl, and long-tailed skua). In this paper we conduct a comprehensive investigation of the complex dynamics within a predator-prey model that incorporates both specialist and generalist predators, alongside constant prey emigration, using a dynamical systems approach. We identify several key bifurcations related to equilibrium and limit cycles, including saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. The presence of cusps in equilibria and limit cycles reveals mechanisms that can lead to three coexisting equilibria or limit cycles, as well as potential extinction for both populations. Additionally, we explore the Fennoscandia phenomenon by examining the interactions between specialist and generalist predators within this framework. To our knowledge, we are the first to fully uncover the real generating mechanism for three limit cycles near a codimension-2 cusp region in the context of lemming dynamics; notably, these limit cycles may bifurcate from different Hopf bifurcation points rather than originating from a single point. Our findings indicate that an increase in the presence of constant lemming emigration can expand the bistable region and resilience of the predator-prey system, accompanied by hysteresis phenomena. This suggests that these factors may enhance the robustness of the predator-prey dynamics.

We consider a model for the dynamics of active cells interacting with their quiescent counterparts under the influence of acidity characterized by proton concentration. The active cells perform nonlinear diffusion and infer proliferation or decay, according to the strength of spatially nonlocal intra- and interspecific interactions. The two cell phenotypes are interchangeable, the transitions depending on the environmental acidity. We prove global existence of a weak solution to the considered PDE-ODE-PDE system and perform numerical simulations in 1D to informally investigate boundedness and patterning behavior in dependence of the system's parameters and kernels involved in the nonlocal terms.

In this note, we study the point symmetries and conservation laws of the conformable fractional one-dimensional heat equation, and provide (1) Lie algebra of infinitesimal point symmetries, (2) an optimal system of one-dimensional subalgebras of this Lie algebra, and (3) the characteristic of conservation laws for this equation. Based on these, we discuss the conformable heat kernel of the conformable heat equation, and the thermodynamic phenomena of the material satisfying the conformable heat equation. From our result, one can also see how the orders of the conformable fractional derivatives affect the amount of symmetry of the equation. As an application, we calculate the group invariant solutions corresponding to the optimal system of one-dimensional subalgebras mentioned above.

This paper investigates the influence of inter-regional population movement on the transmission of infectious diseases through the development of a two-patch SIS epidemic model incorporating stochastic mobility. The theoretical analysis reveals that the long-term behavior of the disease, whether it dies out or persists, is fully characterized by the Lyapunov exponent, $ \lambda $, which quantifies the asymptotic growth rate of the infected population. Specifically, the disease will eventually vanish if $ \lambda < 0 $, whereas it will persist with a unique stationary distribution when $ \lambda > 0 $. A significant methodological challenge arises in establishing the weak convergence of the invariant measure from the stochastic system to its associated two-dimensional boundary system. Owing to the complexity inherent in the boundary system, obtaining the density function of its invariant measure via the Fokker–Planck equation proves to be highly nontrivial. This issue is overcome by employing limiting measures in conjunction with Prokhorov's theorem, thereby offering fresh insights and robust methodologies. Complementary numerical simulations further elucidate the effects of both population movement and environmental fluctuations on disease transmission dynamics.

In this work, we first establish a stability theorem for traveling wave solutions in a class of non-cooperative reaction-diffusion systems featuring nonlocal dispersal of general diffusivities. Our stability condition requires that the initial perturbation leads to a bounded and integrable weighted relative entropy functional. Finally, we demonstrate the applicability of our main theorem to the stability of traveling waves in several non-cooperative models from ecology.

In this paper, we focus on the global dynamics and asymptotic spreading speed for a diffusive Lotka-Volterra type competition system with free boundary and time delay. The well-posedness of solution is established, and some sufficient conditions for spreading and vanishing are derived by giving a suitable initial value compatibility condition. When spreading persists, the asymptotic spreading speed for the invasive species is given by studying an associated semi-wave problem. It is shown that time delay can slow down the spreading speed.

We investigate traveling wave solutions in the two-species reaction-diffusion Lotka–Volterra competition system under weak competition. For the strict weak competition regime $ (b<a<1/c, \, d>0) $, we construct refined upper and lower solutions combined with the Schauder fixed point theorem to establish the existence of traveling waves for all wave speeds $ s\geq s^*: = \max\{2, 2\sqrt{ad}\} $, and provide verifiable sufficient conditions for the emergence of non-monotone waves. Such conditions for non-monotone waves have not been explicitly addressed in previous studies. It is interesting to point out that our result for non-monotone waves also holds for the critical speed case $ s = s^* $. In addition, in the critical strong-weak competition case $ (b<a = 1/c, \, d>0) $, we rigorously prove, for the first time, the existence of front-pulse traveling waves.

In this paper, we use energy estimates to prove inviscid damping and enhanced dissipation for the two dimensional damped linear incompressible MHD equations with Navier boundary conditions in three distinct geometries, namely the whole plane $ \mathbb{R}\times\mathbb{R} $, the half-plane $ \mathbb{R}\times[0, \infty) $ and the infinite channel $ \mathbb{R}\times[-1, 1] $. This study extends the linear results of [3].

In this paper, we study an intra- and interspecific competition system in an unstirred chemostat. We first investigate the local and global stability of two semi-trivial solutions. Then, it follows from monotone dynamical system theory and linear eigenvalue theory that we obtain a relatively clear dynamic classification of this system in the $ \beta_{1}-\beta_{2} $ plane, where the positive constants $ \beta_{i}(i = 1,2) $ represent the maximal growth rate of two species, respectively. The results show that there exist eight curves in the $ \beta_{1}-\beta_{2} $ plane which divide the dynamics of this system into coexistence of two species, extinction of two species, and competition exclusion. Biologically speaking, compared with the classic competitive model, intra- and interspecific competition greatly enriches the competitive outcomes of microorganisms.