We introduce a general class of transport distances $\mathrm {WB}_{\Lambda }$ over the space of positive semi-definite matrix-valued Radon measures $\mathcal {M}(\Omega, \mathbb {S}_+^n)$, called the weighted Wasserstein–Bures distance. Such a distance is defined via a generalised Benamou–Brenier formulation with a weighted action functional and an abstract matricial continuity equation, which leads to a convex optimisation problem. Some recently proposed models, including the Kantorovich–Bures distance and the Wasserstein–Fisher–Rao distance, can naturally fit into ours. We give a complete characterisation of the minimiser and explore the topological and geometrical properties of the space $(\mathcal {M}(\Omega, \mathbb {S}_+^n),\mathrm {WB}_{\Lambda })$. In particular, we show that $(\mathcal {M}(\Omega, \mathbb {S}_+^n),\mathrm {WB}_{\Lambda })$ is a complete geodesic space and exhibits a conic structure.

Geometric parameters in general and curvature in particular play a fundamental role in our understanding of the structure and functioning of real-world networks. Here, the discretisation of the Ricci curvature proposed by Forman is adapted to capture the global influence of the network topology on individual edges of a graph. This is implemented mathematically by assigning communicability distances to edges in the Forman–Ricci definition of curvature. We study analytically both the edge communicability curvature and the global graph curvature and give mathematical characterisations of them. The Forman–Ricci communicability curvature is interpreted ‘physically‘ on the basis of a non-conservative diffusion process taking place on the graph. We then solve analytically a toy model that allows us to understand the fundamental differences between edges with positive and negative Forman–Ricci communicability curvature. We complete the work by analysing three examples of applications of this new graph-theoretic invariant on real-world networks: (i) the network of airport flight connections in the USA, (ii) the neuronal network of the worm Caenorhabditis elegans and (iii) the collaboration network of authors in computational geometry, where we strengthen the many potentials of this new measure for the analysis of complex systems.

This paper is concerned with a predator–prey system with hunting cooperation and prey-taxis under homogeneous Neumann boundary conditions. We establish the existence of globally bounded solutions in two dimensions. In three or higher dimensions, the global boundedness of solutions is obtained for the small prey-tactic coefficient. By using hunting cooperation and prey species diffusion as bifurcation parameters, we conduct linear stability analysis and find that both hunting cooperation and prey species diffusion can drive the instability to induce Hopf, Turing and Turing–Hopf bifurcations in appropriate parameter regimes. It is also found that prey-taxis is a factor stabilizing the positive constant steady state. We use numerical simulations to illustrate various spatiotemporal patterns arising from the abovementioned bifurcations including spatially homogeneous and inhomogeneous time-periodic patterns, stationary spatial patterns and chaotic fluctuations.

Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source

\begin{align*} |x|^{-2}\partial _tu=\Delta u+|x|^{\sigma }u^p, \quad (x,t)\in {\mathbb {R}}^N\times (0,T), \end{align*}

$p\gt 1$

$\sigma \ge -2$

$t\to \infty$

$\sigma =-2$

This paper is focused on the existence and uniqueness of nonconstant steady states in a reaction–diffusion–ODE system, which models the predator–prey interaction with Holling-II functional response. Firstly, we aim to study the occurrence of regular stationary solutions through the application of bifurcation theory. Subsequently, by a generalized mountain pass lemma, we successfully demonstrate the existence of steady states with jump discontinuity. Furthermore, the structure of stationary solutions within a one-dimensional domain is investigated and a variety of steady-state solutions are built, which may exhibit monotonicity or symmetry. In the end, we create heterogeneous equilibrium states close to a constant equilibrium state using bifurcation theory and examine their stability.

In this paper, we report the spatiotemporal dynamics of an intraguild predation (IGP)-type predator–prey model incorporating harvesting and prey-taxis. We first discuss the local and global existence of the classical solutions in N-dimensional space. It is found that the model has a global classical solution when controlling the prey-taxis coefficient in a certain range. Thereafter, we focus on the existence of the steady-state bifurcation. Moreover, we theoretically investigate the properties of the bifurcating solution near the steady-state bifurcation critical threshold. As a consequence, the spatial pattern formation of this model can be theoretically confirmed. Importantly, by means of rigorous theoretical derivation, we provide discriminant criteria on the stability of the bifurcating solution. Finally, the complicated patterns are numerically displayed. It is demonstrated that the harvesting and prey-taxis significantly affect the pattern formation of this IGP-type predator–prey model. Our main results of this paper reveal that (i) The repulsive prey-taxis could destabilize the spatial homogeneity, while the attractive prey-taxis effect and self-diffusion will stabilize the spatial homogeneity of this model. (ii) Numerical results suggest that over-harvesting for prey or predators is not advisable, it can lead to an ecological imbalance due to a significant reduction in population numbers. However, harvesting within a certain range is a feasible approach.