Fig. 2: Mathematical model of dispersal across
two sub-habitats of differing environmental variability. The
model is divided into two sub-habitats: 1 and 2. Each sub-habitat has a
starting population of either the same or different sizes. The arrows
branch-off to demonstrate the fate of the offspring: a proportion will
not disperse, and the rest will. Of those that disperse, some will fail
to disperse, and some will die before reaching the other sub-habitat. In
each generation, sub-habitat 2 remains constant in the number of
offspring produced and therefore, the dispersing to non-dispersing ratio
is constant. Sub-habitat 1, on the other hand, experiences environmental
variability.
Here, we constructed a matrix model to describe dispersal in two
sub-habitats: 1 and 2 (Fig. 2). The model gives the number of
individuals that will be in each sub-habitat in the following
generation. The model describes an annual plant, and so there will be no
overlapping generations, and that dispersal is not a cognitive decision,
but it can be informed by environmental clues. The number of
offspring/fecundity of each plant can be the same or different in each
sub-habitat. There are two variables for dispersal, one for each
habitat. This allows for the rate at which the offspring disperse from
either sub-habitat to be altered (Fig. 2). The diagramatic model in
figure 2 is representative of the matrix model used in this paper:
\(N^{{}^{\prime}}=\par
\begin{bmatrix}1-d_{1}(1-c)\left(S_{1}+\left(\frac{v}{f}\right)\right)&\left(1-\mu\right)d_{2}S_{2}\left(1-c\right)\\
\left(1-\mu\right)d_{1}\left(S_{1}+\left(\frac{v}{f}\right)\right)\left(1-c\right)&\left(1-d_{2}\left(1-c\right)\right)S_{2}\\
\end{bmatrix}N\). (1)
The model is divided into sub-habitats 1 and 2, each starting with a
population of size Nx. The arrows branch-off to
demonstrate the fate of the offspring (S1): a proportion
will not disperse (1-dx), and the rest will
(dx). Of those that disperse, some will fail to disperse
and remain in the native sub-habitat (1-c), where they will attempt
leave the native sub-habitat but remain in the native sub-habitat, and
some will die before reaching the other sub-habitat (µ). In each
generation, sub-habitat 2 remains constant in the number of offspring
produced and therefore, the dispersing to non-dispersing ratio is
constant. Sub-habitat 1, on the other hand, experiences environmental
variability, with a severity (v), which fluctuates the number of
offspring produced (fecundity) depending on whether the environment is
good or bad in that year, and the frequency in which there are good
years (\(\frac{1}{f}\)), and so the number of offspring differ
year-on-year. The matrix demonstrated above is for good years. In bad
years, the fecundity is\(\left(S_{1}-\left(1-\left(\frac{1}{f}\right)\right)v\right).\)This is represented by the width of the arrows. The pattern of
variability is changed for each simulation. The fitness of the
population was measured as the long-term average population growth rate.
We assume that sub-habitat 2 has a constant, homogenous environment.
Sub-habitat 1 experiences a range of differently variable environments.
We define environmental variability by 2 variables: frequency and
severity of “bad years”. Productive years, when fecundity is high, are
noted as “good years”. Frequency is measured by the number of bad
years between good years. In a bad year, only a percentage of the
offspring produced in a good year will survive. This percentage is
determined by the severity variable.