Fronts from two-dimensional dispersal kernels: Beyond the nonoverlapping-generations model
Daniel R. Amor and Joaquim Fort
Departament de Física, Universitat de Girona, Girona, 17071 Catalonia, Spain
Received 30 July 2009; revised manuscript received 21 October 2009; published 23 November 2009
Most integrodifference models of biological invasions are based on the nonoverlapping-generations approxi-
mation. However, the effect of multiple reproduction events overlapping generations on the front speed can
be very important especially for species with a long life spam. Only in one-dimensional space has this
approximation been relaxed previously, although almost all biological invasions take place in two dimensions.
Here we present a model that takes into account the overlapping generations effect or, more generally, the
stage structure of the population, and we analyze the main differences with the corresponding nonoverlapping-
generations results.
DOI: 10.1103/PhysRevE.80.051918 PACS numbers: 87.23.Cc, 89.20.-a, 89.75.Fb
I. INTRODUCTION
Reaction-diffusion and reaction-dispersal fronts have
many applications in physical, biological, and cross-
disciplinary systems 1–3, e.g., virus infection fronts 4,5,
combustion fronts 6,7, human population fronts 8,9, etc.
Motivated by Reid’s paradox of rapid tree range expansions,
recently we have proposed a framework which is useful in
two-dimensional 2D space under the assumption of non-
overlapping generations 10. Modeling forest postglacial re-
colonization fronts by using single-kernel reaction-dispersal
assumptions results in the underestimates of the observed
speeds this disagreement is known as Reid’s paradox. In
order to better predict such speeds, our recent work intro-
duced several-component kernels with characteristic dis-
tances differing several orders of magnitude10. In this
way, long-distance dispersal even if occurring infrequently
makes it possible to predict speeds of the right order of mag-
nitude, as observed from postglacial tree recolonization
fronts.
However, previous work in two dimensions did not take
the age structure of tree populations into account. Indeed,
trees reproduce every year and not only once in their life-
time, so generations clearly overlap. Therefore, here we will
extend the 2D model 10 to overlapping generations. We
shall show that the corrections relative to the nonoverlap-
ping approximation are relevant, which justifies the impor-
tance of our model. Previously, overlapping-generation mod-
els have been only developed in one dimension 11–18. Our
model is relevant not only to tree species, but can be applied
to compute front speeds also in other biophysical and physi-
cal systems in which the reproductive or reactive process
happens more than once for each individual or particle.
II. EVOLUTION EQUATION
A. Nonstructured populations in two dimensions
(continuous space random walk)
Integrodifference equations have been widely used to
model biophysical and cross-disciplinary reaction-dispersion
phenomena. For example, for the case of trees population
dispersion seed dispersal takes place just after reproduction
seed production. Thus the evolution of a nonstructured
population in a 2D space is driven by the well-known inte-
grodifference equation 19
px, y, t + T = R
0
-
+
-
+
px +
x
, y +
y
, t
x
,
y
d
x
d
y
, 1
where px , y , t + T is the population density at the location
x , y and time t + T. Recently we have argued that this evo-
lution equation is also relevant to other biological species
besides trees, e.g., humans 20. However, for the sake of
definiteness and clarity, in this paper we will consider trees
in our explanations. The time interval T is that between two
subsequent dispersal events or “jumps” in the nonstructured
model, T is one generation, i.e., the mean age of trees when
they begin to produce seeds. R
0
is the net reproductive rate
number of seeds per parent tree and year which survive into
an adult tree. Equation 1 is the nonoverlapping-
generations model. It is worth to stress that in this model, the
net reproductive rate per year is always used for R
0
19. The
dispersal kernel
x
,
y
is the probability per unit area that
a particle that a seed falling from a parent tree located at
x +
x
, y +
y
, t reaches the ground at x , y , t + T. Strictly,
Eq. 1 is valid only at sufficiently low values of the popu-
lation density p, because there is a maximum saturation den-
sity above which net reproduction vanishes see Eq. 9 in
Ref. 20; however, this point does not affect the computa-
tion of front speeds because, as we shall see below, such
computations can be performed at low values of p.
Equation 1 is a continuous space random-walk CSRW
equation in two dimensions. It is just an integration over all
possible jumps, which takes into account the probability of
each possible jump dispersal kernel
x
,
y
as well as
the productivity of new individuals net reproduction rate
R
0
.
Let us first summarize some previous results, and we will
then extend them to more general situations. The speed of
fronts evolving according to Eq. 1 can be obtained under
some general assumptions, as follows 10,21. We assume
that R
0
1, that the initial population density has bounded
support i.e., that px , y , t vanishes outside a finite region,
and that for t → the front becomes approximately planar at
scales much larger than that of individual dispersal events.
PHYSICAL REVIEW E 80, 051918 2009
1539-3755/2009/805/0519189 ©2009 The American Physical Society 051918-1