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Critical Patch Sizes and Stability in
Reaction-Diffusion Equations
18.306 Term Paper
Norman Cao
May 14, 2014
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1 Introduction
Reaction-diffusion equations are used to model a wide variety of population dynamics in physical systems. They can predict behaviors of systems where spatial
interactions are important to the qualitative behavior of systems of coupled reactions.
One question that can be asked is: what role do spatial interactions play in the
stability of the system? Under what circumstances do spatial interactions change
the stability of systems from what is expected from the reaction system alone? As
an example, the reaction-diffusion PDE
∂ N (x,t)
∂t
= D∂
2 N (x,t)
∂x2
+ γN (x, t) on 0 < x < L
(1)
N (0, t) = N (L, t) = 0
for D, γ > 0 has N (x, t) = 0 as a globally stable and attracting solution if
p
L < L∗ = π D/γ and no globally stable and attracting solutions otherwise (see
Skellam [11]). However, the corresponding reaction dNdt(t) = γN (t) never has any
globally stable and attracting solutions. In terms of population dynamics, this
question can be framed in the context of questions of extinction and persistence of
populations. Is an area big enough to sustain a population, or will that population
go extinct?
An overview of different types of equations describing reaction-diffusion is given
in §2. The effect of boundary conditions is reviewed in more detail and interpreted
physically in terms of critical patch sizes in §3. Finally, the connection between
semi-discrete and continuous reaction-diffusion equations is explored in §4.
2 Types of Reaction-Diffusion Equations
The governing equations for reaction-diffusion can be either discrete or continuous
in either time or space. The reaction and diffusion laws can vary either in time
or in space. These problems can be considerably harder to find analytical results
for, so this paper will focus on reaction-diffusion which are either homogeneous in
space and time, or have simple discrete variation in time and space.
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2.1 Continuous Models
The continuous time, continuous space case is described by a partial differential
equation (PDE), along with conditions on the boundary. For N different species,
u = u(x, t) = {u1 (x, t), u2 (x, t), . . . , uN (x, t)} is a vector-valued function representing population density of different species over a spatial domain x ∈ Ω for a
given time t:
∂ui
= µi ∆ui + ri (u)ui
(2)
∂t
where µi determines the rate of diffusion, and ri (u) gives the rate of reaction.
The fully continuous case is useful when populations are large and when the spatial
interaction scale between individuals is small compared to the length scale of the
domain.
Turing [12] established the use of PDEs in population dynamics by showing that
a simple two-component reaction-diffusion PDE could exhibit complex behavior
like standing waves or spots starting from small perturbations.
2.2 Semi-Discrete Models
The continuous time, discrete space (CTDS) case is described by a system of
ordinary differential equations (ODEs). For M patches and N species, there are
M × N functions of time uji (t) describing the population density of the i-th species
on the j-th patch. With the vector of populations for each patch is given by
uj (t) = uj1 (t), uj2 (t), . . . , ujN (t):
X kj
duji
= ri (u)ui +
Di (uki − uji )
dt
k6=j
(3)
where Dijk is the diffusion coefficient from patch k to patch j for species i and
the rate function defined similar to the fully continuous case. The discrete space,
continuous time case can be used for modeling a number of spatially isolated
populations, such as populations on islands.
The CTDS models are the most well-studied, since they are a natural extension
of simple ODE models like the Lotka-Volterra Equations:
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dx
dt
dy
dt
= x(α − βy)
= −y(γ − δx)
(4)
or for more general population systems as used by Hastings [5]:
dxi (t)
= xi (t) qi − fi (x1 , . . . , xN ))
(5)
dt
there are a large class of problems of the form 5 for which the global stability is
well understood. See Hastings [5] and Allen [2].
The discrete time, continuous space case (DTCS) can be described by an integrodifference equation. In this case, the population (or population density) at each
discrete time tj , u(x, tj ) can be calculated as the convolution of some dispersal
kernel k(x, y) = k(x − y) with the growth function fi (u(x, tj−1 )):
Z
k(x, y)fi (u(x, tj−1 )) dV + gi (u).
ui (x, tj ) =
(6)
Ω
There can also be additional coupling between populations, as expressed by
gi (u). This case is useful for modeling populations with non-overlapping generations, such as annual plants species or seasonal outbreaks.
3 Boundaries and Critical Patch Sizes
Boundary conditions are necessary to form well-posed PDE problems and as seen
in the example equation 1, can have a dramatic effect on the qualitative behavior
of solutions.
3.1 Boundary Effects
The boundary effects usually associated with PDEs are from boundary conditions.
The three most common types of boundary conditions for a population u for
ˆ are Dirichlet (D), Robin (R), and Neumann (N)
x ∈ ∂Ω with normal vector n
boundary conditions.
u(x, t) = h(x, t)
(D)
ˆ = h(x, t)
u(x, t) + β∇u(x, t) · n
(R)
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ˆ = h(x, t)
∇u(x, t) · n
(N)
The Dirichlet condition can be interpreted as forcibly maintaining the population density at the boundary to be a certain value, either by killing or introducing
individuals as necessary. The Neumann condition can be interpreted as forcibly
maintaining a certain population flux into or out of the domain. The Robin condition can be interpreted as a mixed situation where individuals are put into or
removed from the boundary proportional to the difference between some fixed
number of individuals and the number of individuals at the boundary.
Most commonly, these conditions are seen with h = 0, called the homogeneous
case. However, more complex boundary effects can be created from the interaction of multiple species. Cantrell [3] introduces various predation models at the
boundary with different length scales and shows that the critical patch size can be
increased, and that increase is related to the length scale of the predation.
An intuitive explanation for critical patch sizes is that Dirichlet and Robin
boundary conditions introduce mortality or flux out of domain at the boundaries.
As the patch size increases, the ratio between the boundary size and domain size
decreases, so the boundary has less of an effect on the bulk behavior inside the
domain. See Fagan [4] for a more thorough exploration of possible boundary
effects.
3.2 Linearization Techniques
One of the key points in Holmes [7] is that close to the critical patch size, population
density is small so as long as growth rates do not change sign at small densities,
then the linear effects can be used to determine critical patch sizes. Thus, for
single-population systems
∂u
= D∆u + f (u) ≈ D∆u + uf 0 (0)
(7)
∂t
p
the critical size must be related to the only length scale in the problem, D/f 0 (0).
The constant of proportionality is then determined by the geometry of the domain.
The technique used in Allen [1], Allen [2], and Latore [9] is essentially a linearization technique. In these papers, a critical condition is derived for a semi-discrete
system by means of comparison between a non-linear problem to a solvable linear
problem.
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In particular, for a CTDS system, for certain classes of the rate function ri
(namely bounded), it is possible to construct a new set of differential equations
X kj
duji
dwij
dwij
= Ri wi +
<
Di (wik − wij ) s.t.
dt
dt
dt
k6=j
(8)
Since ui > 0, wi > ui . Then, it is possible to use matrix eigenvalue methods to
determine whether or not wi (t) → 0 as t → ∞. This provides a sufficient condition
for determining extinction.
4 Semi-Discrete to Continuous
Some of the results presented in the previous sections relate to semi-discrete systems where coupled ODEs represent continuous time evolution of population densities in discrete patches. A natural question to ask is whether or not these results
can be generalized to the fully continuous case of a PDE. Some preliminary results
pointing in the right direction are presented in this section.
4.1 Graph Laplacian to Continuous Laplacian
A natural way to extend results of the semi-discrete system to a continuous system
would be to increase the number of patches. One way to formalize the diffusion
between patches, assuming symmetric diffusion between patches, is to use the
notion of a graph. The patches correspond to vertices, and edges correspond to
non-zero diffusion coefficients.
In this setting, for fixed i the matrix (Li )jk = Dijk corresponds to a general
Graph Laplacian. A function f (x) can then be appromixated from the values
at the vertices by using a kernel of bandwidth h. Recent work by Hein [6] and
[10] shows that for randomly sampled patches from a continuous manifold, as
the sample size increases and the bandwidth of the kernel decreases, the Graph
Laplacian Li converges almost surely to a weighted Laplace-Beltrami Operator,
which in Euclidean space is equivalent to the Laplacian.
This connection makes it explicit that increasing patch number while changing
other parameters in such a way to preserve length, time, and population scales
may be a fruitful way to explore global stability of reaction-diffusion PDEs from
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the global stability of similar ODEs. Kuniya [8] provides numerical evidence that
the discretization approach can be useful in global stability analysis with an SIR
epidemic model.
References
[1] Linda JS Allen. Persistence and extinction in lotka-volterra reaction-diffusion
equations. Mathematical Biosciences, 65(1):1–12, 1983.
[2] Linda JS Allen. Persistence, extinction, and critical patch number for island
populations. Journal of mathematical biology, 24(6):617–625, 1987.
[3] Robert Stephen Cantrell, Chris Cosner, and William F Fagan. Habitat edges
and predator–prey interactions: effects on critical patch size. Mathematical
biosciences, 175(1):31–55, 2002.
[4] William F Fagan, Robert Stephen Cantrell, and Chris Cosner. How habitat
edges change species interactions. The American Naturalist, 153(2):165–182,
1999.
[5] Alan Hastings. Global stability in lotka-volterra systems with diffusion. Journal of Mathematical Biology, 6(2):163–168, 1978.
[6] Matthias Hein, Jean-Yves Audibert, and Ulrike Von Luxburg. From graphs to
manifolds–weak and strong pointwise consistency of graph laplacians. Learning theory, pages 470–485, 2005.
[7] EE Holmes, MA Lewis, JE Banks, and RR Veit. Partial differential equations
in ecology: spatial interactions and population dynamics. Ecology, pages 17–
29, 1994.
[8] Toshikazu Kuniya. Global stability analysis with a discretization approach for
an age-structured multigroup sir epidemic model. Nonlinear Analysis: Real
World Applications, 12(5):2640–2655, 2011.
[9] J Latore, P Gould, and AM Mortimer. Spatial dynamics and critical patch
size of annual plant populations. Journal of theoretical biology, 190(3):277–
285, 1998.
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[10] Amit Singer. From graph to manifold laplacian: The convergence rate. Applied
and Computational Harmonic Analysis, 21(1):128–134, 2006.
[11] JG Skellam. Random dispersal in theoretical populations. Biometrika, pages
196–218, 1951.
[12] Alan Mathison Turing. The chemical basis of morphogenesis. Series B, Biological Sciences, 237(641):37–72, 1952.
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