In this paper, we investigate the global boundedness, asymptotic stability and pattern formation of predator–prey systems with density-dependent prey-taxis in a two-dimensional bounded domain with Neumann boundary conditions, where the coefficients of motility (diffusiq‘dfdon) and mobility (prey-taxis) of the predator are correlated through a prey density-dependent motility function. We establish the existence of classical solutions with uniform-in time bound and the global stability of the spatially homogeneous prey-only steady states and coexistence steady states under certain conditions on parameters by constructing Lyapunov functionals. With numerical simulations, we further demonstrate that spatially homogeneous time-periodic patterns, stationary spatially inhomogeneous patterns and chaotic spatio-temporal patterns are all possible for the parameters outside the stability regime. We also find from numerical simulations that the temporal dynamics between linearised system and nonlinear systems are quite different, and the prey density-dependent motility function can trigger the pattern formation.

The current emphasis of schistosomiasis control is placed on preventive chemotherapy using praziquantel. However, reinfection may occur rapidly in the absence of complementary interventions. Recent studies from Senegal suggest that predatory prawns might feed on intermediate host snails and thus impact on schistosomiasis transmission. We designed a study with four repeated cross-sectional surveys pertaining to prawns and snails, coupled with a single cross-sectional parasitological survey among humans. We assessed for potential associations between the presence/density of prawns and snails and correlation with Schistosoma infection in a composite sample of school-aged children and adults. The study was carried out between October 2015 and December 2016 in 24 villages located near the Agnéby and Mé coastal river systems in south-eastern Côte d'Ivoire. At each site, snails and prawns were collected, and in each village, 150 individuals were subjected to stool and urine examination for the diagnosis of Schistosoma mansoni and S. haematobium. We found peaks of relative abundance of intermediate host snails in the villages of the Agnéby River system, while predatory prawns were predominantly recorded in the Mé River system. A negative association was observed between intermediate host snail densities and riverine prawns; however, no pattern was found between this trend in the predator–prey relationship and the prevalence of human schistosomiasis.

We consider a coupled, logistic predator–prey system with delay. Mainly, by choosing the delay time ${\it\tau}$ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay time ${\it\tau}$ passes some critical values. Based on the normal-form theory and the centre manifold theorem, we also derive formulae to obtain the direction, stability and the period of the bifurcating periodic solution at critical values of ${\it\tau}$. Finally, numerical simulations are investigated to support our theoretical results.

We consider a delay predator–prey system without instantaneous negative feedback and establish some conditions for global attractivity of the positive equilibrium of the system which generalize and improve some of the existing ones. When the system is decoupled, one of the main results reduces to the well-known Wright 3/2 stability condition for the delayed logistic equation.

By using Lebesgue’s dominated convergence theorem and constructing a suitable Lyapunov functional, we study the following almost-periodic Lotka–Volterra model with $M$ predators and $N$ prey of the integro-differential equations

\begin{alignat*}{2} \dot{x}_i(t)\amp=x_i(t)\biggl[b_i(t)-a_{ii}(t)x_i(t)-\sum_{k=1,k\neq i}^{N}a_{ik}(t)\int_{-\infty}^tH_{ik}(t-\sigma)x_k(\sigma)\,\mathrm{d}\sigma\\ \amp\hskip45mm-\sum_{l=1}^{M}c_{il}(t)\int_{-\infty}^tK_{il}(t-\sigma)y_l(\sigma)\,\mathrm{d}\sigma\biggr],\amp\quad i\amp=1,2,\dots,N,\\ \dot{y}_j(t)\amp=y_j(t)\biggl[-r_j(t)-e_{jj}(t)y_j(t) +\sum_{k=1}^{N}d_{jk}(t)\int_{-\infty}^tP_{jk}(t-\sigma)x_k(\sigma)\,\mathrm{d}\sigma \\ \amp\hskip45mm-\sum_{l=1,l\neq j}^{M} e_{jl}(t)\int_{-\infty}^tQ_{jl}(t-\sigma)y_l(\sigma)\,\mathrm{d}\sigma\biggr],\amp\quad j\amp=1,2,\dots,M. \end{alignat*}

Some sufficient conditions are obtained for the existence of a unique almost-periodic solution of this model. Several examples show that the obtained criteria are new, general and easily verifiable.

Recently, ratio-dependent predator–prey systems have been regarded by some researchers as being more appropriate forpredator–prey interactions where predation involves serious searching processes. Due to the fact that every populationgoes through some distinct life stages in real-life, one often introduces time delays in the variables being modelled.The presence of time delay often greatly complicates the analytical study of such models. In this paper, thequalitative behaviour of a class of ratio-dependent predator–prey systems with delay at the equilibrium in theinterior of the first quadrant is studied. It is shown that the interior equilibrium cannot be absolutely stable andthere exist non-trivial periodic solutions for the model. Moreover, by choosing delay $\tau$ as the bifurcationparameter we study the Hopf bifurcation and the stability of the periodic solutions.

AMS 2000 Mathematics subject classification: Primary 34C25; 92D25. Secondary 58F14