The collective effect of finite-sized inhomogeneities on the spatial spread of populations in two dimensions

The dynamics of a population expanding into unoccupied habitat has been primarily studied for situations in which growth and dispersal parameters are uniform in space or vary in one dimension. Here, we study the influence of finite-sized individual inhomogeneities and their collective effect on front speed if randomly placed in a two-dimensional habitat. We use an individual-based model to investigate the front dynamics for a region in which dispersal or growth of individuals is reduced to zero (obstacles) or increased above the background (hotspots), respectively. In a regime where front dynamics is determined by a local front speed only, a principle of least time can be employed to predict front speed and shape. The resulting analytical solutions motivate an event-based algorithm illustrating the effects of several obstacles or hotspots. We finally apply the principle of least time to large heterogeneous environments by solving the Eikonal equation numerically. Obstacles lead to a slow-down that is dominated by the number density and width of obstacles, but not by their precise shape. Hotspots result in a speed-up, which we characterize as function of hotspot strength and density. Our findings emphasize the importance of taking the dimensionality of the environment into account.

Invasion front dynamics in disordered environments

Invasion occurs in environments that are normally spatially disordered, however, the effect of such a randomness on the dynamics of the invasion front has remained less understood. Here, we study Fisher’s equation in disordered environments both analytically and numerically. Using the Effective Medium Approximation, we show that disorder slows down invasion velocity and for ensemble average of invasion velocity in disordered environment we have $$\bar{v}=v_0 (1-|\xi |^2/6)$$ v ¯ = v 0 ( 1 - | ξ | 2 / 6 ) where $$|\xi |$$ | ξ | is the amplitude of disorder and $$v_0$$ v 0 is the invasion velocity in the corresponding homogeneous environment given by $$v_0=2\sqrt{RD_0}$$ v 0 = 2 R D 0 . Additionally, disorder imposes fluctuations on the invasion front. Using a perturbative approach, we show that these fluctuations are Brownian with a diffusion constant of: $$D_{C}= \dfrac{1}{8} \xi ^2\sqrt{RD_0 (1-|\xi |^2/3)}$$ D C = 1 8 ξ 2 R D 0 ( 1 - | ξ | 2 / 3 ) . These findings were approved by numerical analysis. Alongside this continuum model, we use the Stepping Stone Model to check how our findings change when we move from the continuum approach to a discrete approach. Our analysis suggests that individual-based models exhibit inherent fluctuations and the effect of environmental disorder becomes apparent for large disorder intensity and/or high carrying capacities.

Front dynamics in the XY chain after local excitations

We study the time evolution of magnetization and entanglement for initial states with local excitations, created upon the ferromagnetic ground state of the XY chain. For excitations corresponding to a single or two well separated domain walls, the magnetization profile has a simple hydrodynamic limit, which has a standard interpretation in terms of quasiparticles. In contrast, for a spin-flip we obtain an interference term, which has to do with the nonlocality of the excitation in the fermionic basis. Surprisingly, for the single domain wall the hydrodynamic limit of the entropy and magnetization profiles are found to be directly related. Furthermore, the entropy profile is additive for the double domain wall, whereas in case of the spin-flip excitation one has a nontrivial behaviour.