May 20, 2018, Sunday
University of Colorado at Boulder Search A to Z Campus Map University of Colorado at Boulder CU 
Search Links

MBW:Biogeographic studies for the design of Nature Reserves

From MathBio

Jump to: navigation, search


Nature preserves are established throughout the world in order to conserve wildlife habitat and species diversity. While the importance of nature reserves are well accepted, the criteria for choosing the locations and sizes of such reserves to optimally conserve biodiversity is less clear. In his seminal research article, "The island dilemma: lessons of modern biogeographic studies for the design of natural reserves",[1] Jared Diamond proposed that nature reserves are analogous to natural oceanic "islands", and are thus governed by the same processes that influence species diversity for island communities.

The goal of this study was to outline design principles that will maintain the highest species diversity by minimizing species extinction rates. Diamond concludes that, given a certain area to preserve, it is best to preserve a continuous circle of habitat as opposed to smaller, disconnected fragments, although the ideal dimensions of the preserve should be guided by the species of concern and their known ecological preferences.

Project Categorization

  • Mathematics

This model uses differential equations to model extinction rates in a nature reserve. It determines the amount of time it will take for a habitat to achieve equilibrium. Initially, a reserve may hold more species than at equilibrium, but the excess will eventually die out as the system converges to equilibrium. The number of species a reserve can hold is a function of its area and its isolation; the larger the reserve and the closer it is to other reserves, the more species it can hold.

  • Type of Model

Population model. The data used in this model comes from the land-bridge island of Misol (connected to the mainland of New Guinea 10,000 years ago). It estimates the time required to reach equilibrium in the number of species is 17,503 years. Currently, Misol supports 135 species, and its equilibrium is at 65 species.

  • Biological System Studied

This paper examines how natural reserves should be laid out in order to maximize biodiversity. Reserves should be located within close proximity of other reserves and their dimensions should be based on knowledge of the local fauna's preferences.

Biological Theory

Much of the theory underlying Diamond's model was outlined in MacArthur and Wilson's famous book, The Theory of Island Biogeography, which was based largely on data from bird populations. According to the theory, two main factors influence species richness:

  • Distance from the "mainland", or source populations - islands that are closer to the mainland are more accessible by migrants and therefore have higher species diversity than more isolated islands
  • Area of the "island" - larger islands have a greater diversity of habitat types, as well as larger populations, than smaller islands, thereby increasing species richness and lowering extinction rates.

Since the 1960's, numerous studies found a power relationship between species richness and island area, represented as:


where S is species richness, C is a constant, A is the area (in km2), and z is a constant < 1. Based on empirical data, z generally lies within the range of 0.18-0.35 for islands across the world.

Species-area relationship for reptiles and amphibians on selected islands. Revised from MacArthur and Wilson, 1967

Diamond revises this power relationship by equating the constant C to the regional species pool, S_{0}:


This equation was used to determine the ultimate number of species that would be expected in a nature preserve at equilibrium, S_{{eq}}.

Diamond's Model

Presumably, a newly created nature preserve will be smaller than the original habitat and would initially contain more species than expected given the preserve size. Thus, we expect the change in species richness over time, , to be < 0 until equilibrium is reached. The model determines the amount of time it will take for a habitat to achieve equilibrium based on data from islands that were attached to the mainland 10,000 years ago, when sea level was 100-200m lower than present. Model parameters:

  • E, extinction rate (species/yr)
  • I, immigration rate (species/yr)
  • S^{*}, mainland species pool
  • t_{r}, relaxation time, or time required for the species excess to relax to 1/e of the initial excess

The change in species richness with time is determined by the number of immigrant species (I) and the number of species that go extinct (E):


where I=K_{i}(S^{*}-S(t))



At equilibrium,

S_{{eq}}={\frac  {K_{i}S^{*}}{K_{i}+K_{e}}}=0

Assuming that the initial species pool, S(0), exceeds Seq, the rate of species change toward Seq is

dS/dt=I-E=(K_{i}+K_{e})({\frac  {K_{i}S^{*}}{K_{i}+K_{e}}}-S(t))

Thus, the rate of species change is dictated by the excess of species on the 'island'. Setting the boundary condition,S(t)=S(0) at t=0:

{\frac  {S(t)-S_{{eq}}}{S(0)-S{eq}}}=e^{{-t/t_{r}}}

Model Analysis

The model was applied to the land-bridge island of Misol, which was connected to the mainland of New Guinea 10,000 years ago. New Guinea harbors about 325 species, while Misol has 135 species. However, based on its land area (2040 km2), Misol should only be able to accommodate 65 species.

The model parameter values:




result in a relaxation time (tr) of 7600 years. Thus, species relaxation should be 90% complete in 2.303*7600=17,503 years.


An important result of this model is that the rate of species loss will initially be very high and decrease with time. Also, the rate of species loss will be accelerated on smaller patches than larger patches, allowing for measurable decreases in species richness within decades. Finally, the relative diversity of the native habitat and its size will affect the rate and percentage of species loss upon conversion into a nature preserve.

Simulation results of the decrease in species richness over time following isolation/fragmentation
One main issue with this model is that it oversimplifies complex species differences and key rates. Some questionable assumptions in this model are:
Relationship betweeen survival probability and species richness for two bird species. Source: Diamond, 1975
  • immigration and extinction rates remain constant
  • all species are equally likely to and capable of migrating between reserve patches
  • all species have the same survival probability

It quickly becomes apparent that certain species (i.e. birds, insects) would be well suited for these assumptions, however many sessile species that are incapable of migrating through inhospitable "oceans" would suffer disproportionate losses.

The importance of these assumptions is evident when considering the shape, size, and number of habitat patches to construct in order to protect endangered species. These species are generally habitat specialists and therefore typically do not conform to the model's assumptions. Thus, Diamond introduced an empirical method for determining species survival probabilities (incidence function) for species of interest, which shows the probability of a bird species residing on a particular island based on its bird species richness. Typically, a bird species' occurrence on an island reaches a threshold for a particular value of S:

This relationship suggests that certain species are destined for extinction if confined to a preserve smaller than their required habitat area.


A flood of research initiated following the recommendations of Diamond and others, particularly after the IUNC published a World Conservation Strategy that issued guidelines based on these studies. According to Margules et al. 1982,[2] the assumptions and restrictions in applying this model are not being adequately considered and may actually be undermining biodiversity preservation.

In another study, Zimmerman and Bierregaard (1986)[3] found that the clumped distribution of frog mating habitat in the Amazon did not fit the species-area relationship and concluded that preserving habitat based on the species-area relationship would be useless without consideration of species traits (autecology).

Test plot established in the Amazon forest in response to the Island Biogeography debate. From Laurance, 2008

After more than a generation of research, conservation biologists have a much clearer understanding of the important mechanisms that influence species survival in nature reserves (as outlined by Laurance, 2008[4]). For instance, edge effects ("Edge Effects") have a profound influence on species survival in fragmented habitats, but Diamond's island biogeographic model does not account for edge effects. The model is also limited in its application to understanding the impacts of habitat fragmentation on ecosystem processes. For example, it cannot predict changes in ecosystem productivity or nutrient cycling ("Nutrient Cycling").

A more promising - albeit far more complex - approach to modeling species responses to habitat fragmentation may be gained using Species Distribution Models (SDM's). [5]

Hierarchical conceptual diagram for a comprehensive SDM in different habitat types. Source: Guisan and Thuiller 2006

Current models seem to be light-years away from the highly simplified model proposed by Diamond in 1975. However, it seems doubtful that such large strides could be made in the absence of these seminal works and the controversies that they created.

External Links

Hodgson, Moilanen, Wintel, and Thomas, "Habitat area, quality and connectivity: striking the balance for efficient conservation"

"Hodgson et. al" (2011) discuss the factors that affect population viability:

  • habitat area
  • habitat quality
  • the spatial arrangement of habitats
  • properties of the surrounding land around the population

These landscape attributes affect whether certain species can change their distributions in response to climate change. A review of published evidence revealed that changes in habitat area and quality are more significant than changes in spatial arrangement of habitats or attributes of the surrounding land. Even if structural features of the surrounding land have a noticeable effect on species dispersal rates, increases in population viability do not necessarily follow. Large, high-quality habitats are the best locations for colonization and can support populations the best. Given these types of habitats as an alternative, a species's capacity to change its distribution in response to climate change is high. Overall, the authors found that retaining as much of the high-quality, original habitat as possible should be the key focus for conservation, especially in the case of climate change.


  1. Diamond, J.M. "The island dilemma: lessons of modern biogeographic studies for the design of natural reserves" Biological Conservation, 1975, 7:129:
  2. Margules, C., A.J. Higgs, and R.W. Rafe. "Modern biogeographic theory: Are there any lessons for nature reserve designs?" Biological Conservation, 1982, 24:115:
  3. Zimmerman, B.L., and R.O. Bierregaard. "Relevance of the equilibrium theory of island biogeography and species-area relations to conservation with a case from Amazonia" Journal of Biogeography 1986, 13:133.
  4. Laurance W.F. Theory meets reality: How habitat fragmentation research has transcended island biogeographic theory. Biological Conservation 2008, 141:1731,
  5. Guisan, A., and W. Thuiller. Predicting species distribution: offering more than simple habitat models. Ecology Letters 2006, 8:993,