Conceived and designed the experiments: YAR LS. Performed the experiments: YAR LS. Analyzed the data: YAR LS. Contributed reagents/materials/analysis tools: YAR LS. Wrote the paper: YAR LS.
The authors have declared that no competing interests exist.
Synthesising the relationships between complexity, connectivity, and the stability of large biological systems has been a longstanding fundamental quest in theoretical biology and ecology. With the many exciting developments in modern network theory, interest in these issues has recently come to the forefront in a range of multidisciplinary areas. Here we outline a new theoretical analysis specifically relevant for the study of ecological metapopulations focusing primarily on marine systems, where subpopulations are generally connected via larval dispersal. Our work determines the qualitative and quantitative conditions by which dispersal and network structure control the persistence of a set of age-structured patch populations. Mathematical modelling combined with a graph theoretic analysis demonstrates that persistence depends crucially on the topology of cycles in the dispersal network which tend to enhance the effect of larvae “returning home.” Our method clarifies the impact directly due to network structure, but this almost by definition can only be achieved by examining the simplified case in which patches are identical; an assumption that we later relax. The methodology identifies critical migration routes, whose presence are vital to overall stability, and therefore should have high conservation priority. In contrast, “lonely links,” or links in the network that do not participate in a cyclical component, have no impact on persistence and thus have low conservation priority. A number of other intriguing criteria for persistence are derived. Our modelling framework reveals new insights regarding the determinants of persistence, stability, and thresholds in complex metapopulations. In particular, while theoretical arguments have, in the past, suggested that increasing connectivity is a destabilizing feature in complex systems, this is not evident in metapopulation networks where connectivity, cycles, coherency, and heterogeneity all tend to enhance persistence. The results should be of interest for many other scientific contexts that make use of network theory.
Taking advantage of modern network theory, we present a model formulation for determining those factors that control the stability and persistence of complex biological systems. As a case study, we focus on ecological metapopulations, which may be viewed as a set of distinct subpopulations (/sites) that are connected via a dispersal network of arbitrary complexity. Metapopulation persistence is found to depend critically on the topology of cycles, and cyclical components in the connectivity network, because they allow the offspring of the population to eventually “return home” to the sites from which they originated. The methodology identifies critical migration routes, whose presence are vital to overall stability, and are thus of high conservation priority – information that may be of value when designing networks of marine protected areas. In contrast, links that do not participate in a cyclical component have no impact on persistence and thus have low conservation priority. While network theory is highly fashionable in biology, only few studies go deeper than descriptive statistical applications as attempted here. Moreover, the key results are easily extended to other biological contexts (e.g., disease networks), particularly in situations whereby the network controls the dynamics of a complex system.
Theoretical biologists and ecologists have long sought to understand the relationships between complexity, connectivity and the stability of large biological systems
Given the importance of the metapopulation concept for nature conservation, theoretical studies have attempted to gain insights into those factors that make for sustainable populations. Intriguingly, initial research indicated that spatial structure has in fact very little effect, with criteria for metapopulation stability appearing to be identical to the stability conditions for a single patch
Similar to
We begin by considering a population that has
The population has a single equilibrium, the extinction state
Now scaling up, consider a network of
The parameter
A key goal is to examine the effects solely of network architecture on metapopulation persistence. This requires studying the metapopulation with all processes being equal apart from the network structure. We thus initially suppose that local patch populations are identical, and all with the same survival and fertility rates
The metapopulation's complex patch dispersal structure may be visualized in terms of a network or graph, with nodes as patches and edges as dispersal links between patches. The network topology is summarised by the adjacency matrix
Again, in order to specifically elucidate the effects of network structure, it is necessary to ensure that the various processes between patches are kept equal. This motivates a relatively simple but nevertheless useful scheme in which the number of juveniles that immigrate to a patch population is on average a fixed proportion,
The connectivity matrix may now be written in the particularly simple form:
The expression (Eqn. 7) for the persistence parameter shows clearly the importance of supply-side ecology with new recruits enhancing the possibility of persistence. A metapopulation, in which all
An exciting outcome of Eqn. 7 is that it opens the door for investigating the effects of network structure, at both fine and coarse-scale levels. Beginning with coarser-scale features, we focus on network topology.
A) Regular, B) Random, and C) Heterogeneous networks, all with mean degree of 2. D) The persistence parameter
It is then natural to ask which topology best enhances persistence and how does the topology's heterogeneity affect the threshold
The three above cases are now treated separately assuming the networks are undirected; that is assuming a dispersal pathway from patch-
In a regular network every patch is connected to
The classical Erdős–Rényi (ER) random network assumes there is a probability
The result can be generalized for heterogeneous random networks having arbitrary degree distribution (
Finer-scale network features also play an important role in determining persistence. The present framework allows exploration of how various network motifs
Firstly, it becomes trivial to show that a metapopulation without any cycles in the dispersal network is unable to persist, as might be expected (see also
Self-recruitment
(A) A complex network with a single simple cycle (red). Eliminating all “lonely links” that do not belong to any cyclic component results in the network shown in (B).
Consider first simple cycles, with no sites receiving recruits directly from two distinct cycles. Directed networks composed of only simple cycles, have the property that
Persistence, is completely determined by the dominant cyclic component. Intriguingly it turns out that valuable information regarding persistence may be obtained by breaking down a network into its nonoverlapping cyclic components. In particular, we are able to show (
The other nine patches (blue, green) may be completely neglected since they do not belong to any cycle and therefore play no role in determining persistence. The characteristic equation for the eigenvalues of the adjacency matrix
It is revealing to return to the earlier examples of
So far, persistence has been discussed in the context of networks of identical patch populations. However this work may be extended to obtain persistence/extinction criteria for metapopulations comprised of nonidentical patch populations thus accommodating cases where age structure varies between patches. For convenience, suppose patch-1 of the metapopulation “dominates” in the sense that it has the highest fertility parameters (
Along similar lines, suppose patch-1 is the weakest in the sense that it has the lowest fertility parameters and lowest survival rates. If the network of
The reproductive number of the patches are such that
It should be noted that the effects of age-structure become prominent with more complex migration schemes, for example, between different age-classes from different patches, and representations of ontogenetic shifts in habitat use that are life history dependent [
Our work has shown the importance of the persistence parameter
Ideally, we also require conditions that establish metapopulation
The random Erdős–Rényi (ER) model which exemplifies a broad range of complex networks shows the existence of important coherency thresholds that must be taken into account. Recall that there is a probability
Taken altogether, the above framework has revealed a number of new insights regarding the determinants of persistence, stability and ecological thresholds in complex metapopulations. We note that a number of results derived here are based on several simplifying assumptions. Thus, at least in the first instance, it was necessary to assume that all patches are identical in terms of their structure and dynamics (a condition that was later relaxed). In addition, it was assumed that the rate of dispersal was the same for all pairs of connected patches. In other applications these model limitations may need to be reckoned with. However, for our purposes, only by fixing patches and metapopulation processes equal, is it possible to isolate exactly how different dispersal networks and their topologies govern persistence. This is in fact a major difference between our work here and that of Armsworth
Our analysis was couched both in terms of coarse and fine-scale network features. With regard to the former, at least for random systems, connectivity and heterogeneity in connectivity were found to be two key factors that enhance metapopulation persistence. In terms of more fine-scale features, the existence of critical dispersal routes have been identified, while cycles have been shown to play a prominent functional role, allowing metapopulations to bootstrap themselves into persistence. All of these factors have conservation applications and would translate most readily into principles that aid in the management of marine populations and in the design of networks such as marine protected areas
For a metapopulation of
For the particular case
The persistence parameter χ. Formulation of age-structured metapopulation model and its mathematical stability criteria.
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Cycles. Mathematical analysis of the impact of cycles on network stability.
(1.52 MB PDF)
Nonidentical patches. Analysis of the metapopulation model for nonidentical patches.
(0.38 MB PDF)