A spatially explicit boolean model

The Levins and other spatially implicit models have the limitation that they cannot be used to analyse explicit spatial patterns. For example, colonization is very often distance-dependent; if this feature is introduced into a model, simulation shows that metapopulation persistence is influenced in a different way by habitat loss (smaller h) occurring randomly or nonrandomly in space.

When it comes to which spatially explicit model to use, there are several possible choices, depending on the kind of available data and the goal of the risk assessment to be conducted. Ideally, one could use a geographical information system to detail the biotic and abiotic characteristics of the landscape continuum. Also, more realistic dispersal descriptions can be used, e.g., via dispersal kernels. An intermediate, yet spatially explicit approach consists in dividing the landscape into a finite number of patches, each of them possibly being characterized by its area, resource availability and location in space (so that the relative distances among the patches can be explicitly included). As for the demography of each population hosted in the patches, the most detailed approach would be the one in which the population dynamics is described by birth-death processes like the ones we introduced in the chapter "Extinction risk analysis: demographic and environmental stochasticity". However, here we will assume that the population dynamics of each landscape fragment can be described in a boolean way (patch is empty or occupied). Perhaps, the best known model of this kind is the one introduced by (Hanski and Ovaskainen, 2000).

Consider a number n of patches; term $p_i(t)$ the probability that patch i with $i=1...n$ is occupied, $E_i$ and $C_i$ the relevant extinction and colonization rate, respectively. Then the rate of change of $p_i(t)$ can be described by a Levins-like model of this kind

$$\dot{p_i}(t)=C_i\cdot\left(1-p_i(t)\right)-E_i\cdot p_i(t) .$$ (6.6)

Note, however, that $p_i(t)$ is a probability, not a fraction like the one used in the bona fide Levins model of the previous sections. The extinction and the colonization rate of population i can be specified in the following way. Let $A_i$ be the area of habitat patch i. The extinction rate is assumed to be a function of the inverse of patch area, $E_i=e/A_i$ with e being a positive constant, because large patches tend to have large expected population sizes and because extinction risk scales roughly as the inverse of the expected population size (Hanski (1999)). Notice that the constant e is measured as area per unit time, like the diffusion coefficient D we introduced in the previous chapter. The colonization rate, instead, is given as a sum of contributions from the existing populations. Let $l_{ji}$ be the probability that a propagule released by population j reaches patch i; often, it is specified as

$${l_{ji}} = L \exp \left( { - \alpha {d_{ji}}} \right)$$

where $d_{ji}$ is the distance between the two patches, L is a suitable constant smaller than 1 and $1/\alpha$ is the average dispersal distance. This is an example of an exponential dispersal kernel (but other choices are possible, e.g. a Gaussian kernel, (Clark et al., 1999)). If $l_{ji}$ only depends on distance, then $l_{ji} = l_{ij}$, i.e. the dispersal probability matrix is symmetric. Then the colonization rate is given by

$$C_i=c\sum\limits_{j \ne i} {{l_{ji}}} {A_j}{p_j}(t)$$

because immigration to patch i is expected to increase with the number of neighbouring populations, with their sizes as reflected by the respective patch areas, and with their decreasing distances to the focal patch and increasing probability of occupancy. The constant c is positive and is a probability of colonization per unit area per unit time.

Thus, the dynamics of the metapopulation is given by

$$\dot{p_i}(t)=(c/{A_i})\left[ \sum\limits_{j \ne i} {{m_{ji}}}{p_j}(t)\cdot\left(1-p_i(t)\right)-(e/c)p_i(t) \right].$$ (6.7)

where $m_{ji} = l_{ji}A_iA_j = m_{ij}$ are the elements of a matrix M, which we will term the migration matrix from now on. Model 37 can admit several possible equilibria, but the persistence of the metapopulation can be anyway established by examining the stability of the extinction equilibrium, namely ${p_i} = 0\quad \forall i$. In order to do that, we linearize the model around the zero equilibrium. This amounts to discarding the quadratic terms in the right-hand-side of eq. 37, namely

$$\dot{p_i}(t)=(c/{A_i})\left[ \sum\limits_{j \ne i} {{m_{ji}}}{p_j}(t)-(e/c)p_i(t) \right] .$$ (6.8)

Eq. 38 can be rewritten with matrix notation as

$$\dot p = c{A^{ - 1}}\left( {M - \frac{e}{c}I} \right)p$$

where

$$p = \left[ {\begin{array}{*{20}{c}} {{p_1}}\\ {{p_2}}\\ .\\ .\... ....&.\\ .&.&.&.&.\\ {{m_{1n}}}&{{m_{2n}}}&{{m_{3n}}}&.&0 \end{array}} \right] ,$$

$M$ is the migration matrix, and $I$ is the identity matrix. Since $M$ is a symmetric matrix, its eigenvalues are real. The largest eigenvalue is termed the dominant eigenvalue. (Hanski and Ovaskainen, 2000) have shown that the extinction equilibrium is unstable, so that the metapopulation can persist, if and only if the dominant eigenvalue $\lambda_M$ of $M$ is larger than $e/c$:

$${\lambda _M} > \frac{e}{c}.$$ (6.9)

$\lambda_M$ is measured as a squared area and has been aptly defined metapopulation capacity by (Hanski and Ovaskainen, 2000). Persistence is thus guaranteed (eq. 39) if the metapopulation capacity exceeds the threshold $e/c$. Note that it is possible to recover the result of the simple Levins model. It is sufficient to assume that all patches are equal, have the same unitary area, and there is a rain of propagules (i.e. $l_{ji} = 1/(n-1)$); in that case, as we already know, the condition for persistence is $1 > e/c$. In other words the metapopulation capacity for that model is 1.

The big advantage of a spatially explicit model is that one can specify the location and the area of the habitat patches and simulate the fate of the corresponding metapopulation. It is easy to introduce realistic patterns of habitat loss and environmental catastrophes by manipulating the model parameters. For instance, habitat erosion, that is a proportional reduction of all the habitat patches, can be simulated by assuming that the areas of all the patches are decreased by a certain proportion. Consequently, the migration matrix $M$ changes, its dominant eigenvalue can be recalculated and compared to the threshold $e/c$ to see whether persistence is possible. Of course, simulation of the differential equations 37 allows one to determine also the probability that each specific habitat patch is occupied in both the short and the long term. On the other hand, strategies that aim at the preservation of a species can also be simulated. The establishment of an ecological corridor between patch i and patch j, for instance, can be mimicked by an increase of the relevant dispersal coefficient $l_{ji}$.

The conclusions obtained in this chapter are subject to a number of limitations related to the excessive simplicity of boolean models from which results were derived. The reader should keep this consideration in mind as a guide to planning and managing the conservation of metapopulations. In particular, more sophisticated models (for a better description of either habitat or local demography) should be used if one seeks for solutions to be practically implemented. Sometimes, in fact, boolean models significantly underestimate the extinction risk of metapopulations. An important example in this respect is related to the estimate of the proportion of habitat that must be preserved so that a metapopulation may persist even when part of the habitat is destroyed. The simplicity of the laws derived from the boolean model (34) is indeed very tempting and the risk of using it in decision-making contexts is considerable. Fig. 7A shows the fraction of occupied patches in a hypothetical metapopulation as a function of the preserved habitat fraction $h$ calculated by means of the modified Levins model. As one can note, this relationship is linear and vanishes for $\hat{h}=\frac {e}{c}$. Therefore, following the directions of the boolean model, a decision-maker who has to grant permissions for exploiting the land for other purposes (for example, converting forested fragments into agricultural use, or building resorts in an island archipelago) could be led to believe that a proportion of habitat can be exploited to the limit of $1-\hat{h}=1-\frac {e}{c}$, i.e. exactly the fraction of habitat patches occupied by the species before land-use change. This “rule of thumb” for environmental conservation, although quite popular because of its simplicity, is very gross and should not be used in practice without careful consideration. Fig. 7B displays the proportion of occupied fragments at equilibrium as a function of $h$ in a more sophisticated metapopulation model, which includes local population dynamics and demographic stochasticity, in two different cases. While the dot-and-dashed curve, which refers to a species with low frequency dispersal, rather faithfully reproduces the result that is obtained via the Levins model, the dashed curve (referring to a species with a high frequency dispersal) displays a very different result. In fact, destruction of 40% of the habitat is sufficient to condemn such a metapopulation to extinction even if more than 90% of the fragments were occupied before land-use conversion. Therefore, it is good to remind the reader that, before translating the simple management policies suggested in this chapter into concrete implementation, it is necessary to evaluate the quantitative effects with more appropriate tools than simple boolean models. Available PVA software like RAMAS and VORTEX can help the ecologists to carry out the job.

Figure 7: Fraction of occupied patches at equilibrium in a hypothetical metapopulation as a function of the preserved habitat fraction $h$ calculated by means of different models. (A) Modified Levins model (34); (B) Spatially implicit structured model (Casagrandi and Gatto, 2002b) in the case of high (dashed curve) and low (dash-and-dotted curve) dispersal frequency. Dotted curves correspond to the results that would be obtained by utilizing the simple model by Levins (1969).