Critical Patch Size

We now turn our attention to steady states of reaction-transport systems. We focus first on steady states that arise in RD models on finite domains. Such models are important from an ecological point of view, since they describe population dynamics in island habitats. The main problem consists in determining the critical patch size, i.e., the smallest patch that can minimally sustain a population. As expected intuitively, the critical patch size depends on a number of factors, such as the population dynamics in the patch, on the nature of the boundaries, the patch geometry, and the reproduction kinetics of the population. The first critical patch model was studied by Kierstead and Slobodkin [228] and Skellam [414] and is now called the KISS problem. A significant amount of work has focused on systems with partially hostile boundaries, where individuals can cross the boundary at some times but not at others, or systems where individuals readily cross the boundary but the region outside the patch is partially hostile, or a combination of the above. In this chapter we deal with completely hostile boundaries and calculate the critical patch size for different geometries, reproduction processes, and dynamics. 

Let L be the patch size. Throughout this chapter we consider the RD equation on the interval [0, L] with Dirichlet boundary conditions  

Since the critical patch size corresponds to the borderline between population extinction and survival, the population density p is small and we can linearize (9.1) about p(x) = 0, the state of extinction. If p = 0 is stable, the population goes to extinction. If p = 0 is unstable, the population persists or survives. Near p = 0 

Mendez et al., Reaction-Transport Systems, Springer Series in Synergetics, DOI f O.f007/978-3-642-H443-4 9, Springer-Verlag Berlin Heidelberg 2010 

Since the problem is linear, it is reasonable to look for solutions of the form 

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Spatial distribution across available habitat Critical patch size Habitat requirements... 

The critical patch size is obtained by taking the limit 0 in (9.22) ... 

The method of the normal solution expansion provides a means to obtain the critical patch size if F (0) > 0, the saddle-node bifurcation point if p iff) < 0, and the stability of the steady states. An analytical expression for the steady state can also be obtained, but it is an approximate solution since we truncate the expansion. The normal solution expansion is a well-known method to obtain solutions of the nonlinear Boltzmann equation [180, 359]. It assumes that the distribution function f(jc, v, t), describing the density of atoms or structureless molecules at position x with velocity V at time t depends on time only through the velocity moments /o(jc, t), u(x, t), Pij x, t), i.e., f(jc, V, t) f(jc, V, p(x, t), u(x, t), Pjjix, t)), where p, u, Pij are found by integrating f, fw, and fVjVj, respectively, over the full velocity space. We express the solution of (9.1) in terms of an appropriate complete set of orthogonal spatial basis functions ... 

As is the case with the normal solutions of the Boltzmann equation, the initial conditions may be such that the solution obtained is not accurate for early times (initial slip [180]). The validity of the normal solution method requires that lim (,j o /t>i/ i = 0. We show below that this condition is satisfied. Since we focus on the critical patch size, i.e., the size of the habitat where a transition from population extinction to persistence occurs, we deal with the case where p is a small quantity. We expect therefore that the solution will be well approximated outside an initial time layer by i(t). The equations for j(t), with j = 1, 2,..., can be found by substituting (9.28) into (9.1), multiplying by sin( rx/L), and integrating over the spatial domain. 

In this case there exists a critical patch size. In the following we truncate the expansion in (9.39) at the first order for simplicity, p q> with q> given by (9.44). The nontrivial branches collide at a saddle-node bifurcation point or turning point that... 

Find the critical patch size for the RD equation with Robin boundary conditions... 

Part III focuses on spatial instabilities and patterns. We examine the simplest type of spatial pattern in standard reaction-diffusion systems in Chap. 9, namely patterns in a finite domain where the density vanishes at the boundaries. We discuss methods to determine the smallest domain size that supports a nontrivial steady state, known as the critical patch size in ecology. In Chap. 10, we provide first an overview of the Turing instability in standard reaction-diffusion systems. Then we explore how deviations from standard diffusion, namely transport with inertia and anomalous diffusion, affect the Turing instability. Chapter 11 deals with the effects of temporally or spatially varying diffusivities on the Turing instability in reaction-diffusion systems. We present applications of Turing systems to chemical reactions and biological systems in Chap. 12. Chapter 13 deals with spatial instabilities and patterns in spatially discrete systems, such as diffusively and photochemically coupled reactors. 

Fig. 9.5 (a) Bifurcation diagram for depensation growth for a = 0.1 and D = r =. Solid lines correspond to stable branches and the dotted lines to unstable branches. Symbols depict numerical results. Vertical dashed lines separate the extinction, relative extinction/survival, tind survival regions, (b) Plot of the critical initial density vs the patch size for the relative extinction/survival region... 

In the case of a single patch, the size dependence of the system follows directly from the finite size scaling theory [133]. In particular, the critical point temperature scales with the system size as predicted by the equation... 

The deposition of the first stem of thickness / > lmin is the critical stage of nucleation. However, further stems need to attach next to it before the patch reaches a stable size (see Fig. 3.10). This size can be determined by setting the change in free energy A v of formation of a surface strip possessing v stems to zero ... 

Overall, wettability measurement of small particles is a difficult problem that is further aggravated in the case of heterogeneous surfaces. Some of these problems can result from the presence of patches of different composition in the same particle. It is considered that if these patches are below a critical size of 0.1 mm, the surface is homogeneous regarding its wettability. Several indirect techniques have been developed to measure the surface tension, and thus the wettability of small particles. In these techniques, the surface tensions of the particles are derived from thermodynamic models and include the advancing solidification front or freezing front, sedimentation volume, and particle adhesion techniques [44, 45]. 

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