Competition and niche dynamics are critical for predicting species distributions under future acidification conditions. In this study, we develop a deterministic Lotka-Volterra competition model along a pH gradient using a trait-based approach. By analyzing the combined effects of competition and niche width on the competitive outcomes of aquatic organisms in heterogeneous acidified lakes, our theoretical analysis and numerical simulations yield new insights. We find a tradeoff between competition strength and niche width that facilitates transitions among competitive outcomes, including competitive exclusion, coexistence, and priority effects (bistability). In the absence of dispersal, as pH levels in habitats decrease, distinct transitions in competitive outcomes emerge for competing species with both similar and different niche widths. In the presence of dispersal, we identify two critical curves of niche breadth, which determine whether coexistence or priority effects occur. Generally, species with wider niches are likely to outcompete those with narrower niches. Additionally, inferior competitors with high dispersal rates are more likely to coexist with superior competitors that have lower dispersal rates, suggesting that higher diffusivity can compensate for both weak competition strength and narrow niches. This model provides valuable insights into how niche dynamics and competitive abilities influence competitive outcomes in heterogeneous acidified lakes.

In this paper, we propose a periodic reaction-diffusion model of hospital infection with crowding effects. We introduce the basic reproduction number R0 for this model and show that the infection-free periodic solution is globally asymptotically stable if R0≤1, while the system admits a globally asymptotically stable positive periodic solution if R0>1. We also obtain the asymptotic behavior of the basic reproduction number when the diffusion rates go to infinity and zero, respectively. Further, we numerically study the effects of diffusion rates, the outflow rate by patient crowding and other parameters on R0 with and without seasonality, and found that the neglect of seasonality does overestimate the infection risk and the disease may be well controlled by avoiding overcrowding of patients rather than increasing the mobility of individuals or bacteria.

This paper is devoted to a mathematical model with diffusion and cross-diffusion to describe the interaction between vegetation and soil water. First, the existence of Hopf bifurcation and cross-diffusion-driven Turing instability are discussed. Then, based on the nonlinear analysis, we obtain the exact parameters range for stationary patterns and show the dynamical behavior near Turing bifurcation point. It is found that the model has the properties of gap, strip and spot patterns. Moreover, the small water-uptake ability of vegetation roots promotes the growth of vegetation and the transitions of vegetation pattern. But with the continuous increase of the water-uptake ability of vegetation roots, the local vegetation biomass density increases and the isolation between vegetation patches also increases, which may induce the emergence of desertification. In addition, our results reveal that the water consumption rate induces the transitions of vegetation pattern and prohibits the increase of vegetation biomass density.

In this paper, the dynamic behaviors of tuberculosis in the context of indirect environmental transmission are discussed by establishing the SEIRB epidemic model. The basic reproduction number is computed by employing the next-generation matrix approach. The global stability of disease-free equilibrium and endemic equilibrium is proved by constructing the Lyapunov function and the application of LaSalle’s invariance principle. It shows that when the basic reproduction number is greater than 1, tuberculosis will spread among the population. When the basic reproduction number is less than 1, tuberculosis will disappear. Finally, an optimal control problem is constructed by using the extended model, which reveals the spread of tuberculosis can be effectively controlled by eliminating Mycobacterium tuberculosis in the environment and controlling tuberculosis patients at the same time. Numerical example results show the effectiveness of the optimization strategies.

In this paper, a vegetation–water model with saturated water absorption is considered. The global stability of boundary equilibrium is first obtained. Then the stability and Turing instability of positive equilibria are discussed and we find that the equilibrium with vegetation small is always unstable and if the vegetation diffusion is small or the water diffusion is large, the other positive equilibrium loses its stability and Turing instability occurs. In addition, a priori estimates of nonnegative steady-state solutions is obtained by the maximum principle. Moreover, some detailed qualitative analyses are carried out on the steady-state bifurcations at both simple and double eigenvalues. In particular, we establish the criterion to determine the bifurcation direction. Finally, some dynamics near the bifurcation point are showed numerically and we depict the evolution processes of vegetation patterns under different parameters. Our results show that the parameter 1/p, which represents the conversion of water absorption, has a great impact on the vegetation patterns formation: with the increase of p, the vegetation biomass decreases, and meanwhile it can induce the transition of different pattern structures.

Vegetation patterns can reflect vegetation’s spatial distribution in space and time. The saturated water absorption effect between the soil–water and vegetation plays a crucial role in the vegetation patterns in semi-arid regions. Moreover, vegetation can absorb water through the nonlocal interaction of roots. In this paper, we consider how cross-diffusion and nonlocal delay interactions affect vegetation growth. The conditions under which the vegetation-water model generates the Turing pattern are obtained by mathematical analysis. At the same time, the multiple scales method is applied to obtain the amplitude equations at the critical value of Turing bifurcation, which helps us to derive parameter space more specifically where specific patterns such as strips, hexagons, and the mixture of strip and hexagons will emerge. Various spatial distributions of vegetation in semi-arid areas are qualitatively depicted by numerical simulations. The results show that the nonlocal delay effect enhances vegetation biomass. Therefore, we can take measures to increase the intensity of the nonlocal delay effect to increase vegetation density, which theoretically provides new guidance for vegetation protection and desertification control.

To investigate ocean acidification and Allee effects on the dynamics of a marine predator–prey system, an aquatic diffusive predator–prey model with double Allee effect on prey and pH-dependent capture rate is considered. First, we study the stability of constant steady state solutions using linearized theory. Second, the nonexistence of nonconstant positive steady state solutions is shown for appropriate ranges of parameters. Furthermore, we show the existence of a Hopf bifurcation and derive the direction and stability of the bifurcating periodic solutions. Both theoretical analysis and numerical simulation show that changing predator–prey interaction strengths, due to changing environmental conditions, can fundamentally change the system dynamics, even for apparently small changes in interaction strength. As the interaction strength decreases due to decreasing ocean pH, the system dynamics transition from persistent fluctuations in species abundances (periodic solutions), to stable coexistence, to predator extinct (with stable non-zero prey abundance), suggesting the potential for ocean acidification to decrease the abundance and diversity of marine species by weakening predation rates. Moreover, double Allee effect parameters together determine the stability of periodic solutions when the spatially homogeneous bifurcating periodic solutions exist, and the wavelength becomes longer as the Allee effect increases.

In this paper, we study a cross-infection model with diffusion and incubation period. Firstly, we prove the global attractivity of the infection-free equilibrium and infected equilibrium for the spatially homogeneous system. Secondly, we establish the threshold dynamics for the spatially heterogeneous system in terms of the basic reproduction number \begin{document}$ \mathcal{R}_0 $\end{document}. It turns out that the infection-free steady state is globally attractive if \begin{document}$ \mathcal{R}_0<1 $\end{document}; and the system is uniformly persistent if \begin{document}$ \mathcal{R}_0>1 $\end{document}. Finally, we explore the influence of different diffusion coefficients, spatial heterogeneity of the disease transmission rate and the incubation period on \begin{document}$ \mathcal{R}_0 $\end{document}. Our numerical results show that \begin{document}$ \mathcal{R}_0 $\end{document} are decreasing functions of the diffusion coefficients and the incubation period, respectively, while it is increasing with respect to the spatial heterogeneity.

In order to investigate the role that environmental bacteria played in the dynamics of hospital infections, we propose a cross-infection model with diffusive bacteria in the environment. Firstly, we prove the global existence, uniform boundedness and ultimate boundedness of solutions as well as the existence of a global attractor for the equivalent model. Secondly, we investigate a limiting system to establish the threshold dynamics for the model in terms of the basic reproduction number R0 by using the theories of monotone dynamical systems and chain transitive sets. More precisely, we show that if R0≤1, then the infection-free steady state is globally stable; and if R0>1, then the system has a globally stable endemic steady state. Finally, we use the numerical method to explore the influence of different diffusion coefficients on R0. In the case where the transmission rate is independent of diffusion coefficient, the numerical results indicate that R0 is decreasing with respect to the diffusion rate. In the case where the transmission rate is a function of diffusion coefficient, we find that in a less polluted environment, R0 is a decreasing function with respect to the diffusion rate, which implies that the diffusion of bacteria is beneficial for patients; while in a more polluted environment, R0 may increase with increasing diffusion rate, which means increasing diffusion of bacteria is harmful for the elimination of disease.

We investigate a nonlocal reaction-diffusion-advection model which describes the growth of a single phytoplankton species in a water column with crowding effect. The longtime dynamical behavior of this model and the asymptotic profiles of its positive steady states for small crowding effect and large advection rate are established. The results show that there is a critical death rate such that the phytoplankton species survives if and only if its death rate is less than the critical death rate. In contrast to the model without crowding effect, our results show that the density of the phytoplankton species will have a finite limit rather than go to infinity when the death rate disappears. Furthermore, for large sinking rate, the phytoplankton species concentrates at the bottom of the water column with a finite population density. For large buoyant rate, the phytoplankton species concentrates at the surface of the water column with a finite population density.

In this paper, we propose an SIS-type reaction-diffusion equations, which contains both direct transmission and indirect transmission via free-living and spatially diffusive bacteria/virus in the contaminated environment, motivated by the dynamics of hospital infections. We establish the basic reproduction number R₀ which can act as threshold level to determine whether the disease persists or not. In particular, if R₀<1 then="" the="" disease-free="" equilibrium="" is="" globally="" asymptotically="" stable="" whereas="". For the spatially homogeneous system, we investigate the traveling wave solutions and obtain that there exists a critical wave speed, below which there has no traveling waves, above which the traveling wave solutions may exist for small diffusion coefficient by the geometric singular perturbation method. The finding implies that great spatial transmission leads to an increase in new infection, while large diffusion of bacteria/virus results in the new infection decline for spatially heterogeneous environment.