Copyright © 2002 by the author(s). Published here under license by The Resilience Alliance.
The following is the established format for referencing this article:
Cumming, G. S. 2002. Habitat shape, species invasions, and reserve design: insights
from simple models. Conservation Ecology 6(1): 3. [online] URL: http://www.consecol.org/vol6/iss1/art3/
A version of this article in which text, figures, tables, and appendices are separate files may be found by following this link.
Report, part of Special Feature on Ralf Yorque Memorial Competition 2001
Habitat Shape, Species Invasions, and Reserve Design: Insights from Simple Models
Graeme S. Cumming
University of Florida
ABSTRACT
Species invasions have become a major threat to global biodiversity. We currently lack a general theory of species invasions that allows us to make useful predictions about when and where invasions will occur, whether they will be successful, and whether they will alter ecosystem function in invaded habitats. One line of enquiry in developing such a theory is to focus on the characteristics of successful invaders. A second, complementary approach is to examine habitats of interest more closely and ask how the properties of the habitat that is
being invaded affect the likelihood of invasion success. In this paper, I consider the importance of habitat shape (also termed "habitat topology" or "habitat geometry") as a variable affecting the dispersal and abundance of invasive populations. I use two well-established simulation modeling approaches, namely, a cellular automaton model and a reaction-diffusion model, to mimic species invasions in hypothetical habitats that cover a range of linear, branching, rectangular, and square shapes. The results suggest that invasions in more geometrically complex habitats will occur faster and may ultimately produce a higher abundance of the invasive species. Differences in invasion rates are not a simple consequence of differences in overall connectivity, as shown by a comparison of habitats with identical connectivities but different spatial arrangements of cells. Ultimately, if combined with other modeling approaches, these methods may be useful in generating recommendations for managers about the vulnerability of particular habitats and reserve networks to invasion.
KEY WORDS: cellular automaton model, colonization, connectivity, dispersal, habitat complexity, habitat geometry, habitat shape, landscape ecology, reaction-diffusion model, reserve design, reserve networks, species invasion.
Published: February 13, 2002
INTRODUCTION
Human expansion of the geographical ranges of other species has become a major threat to global biological diversity. Numerous unplanned, uncontrolled experiments have shown how a wide range of invasive species including crayfish, mussels, insects, fish, birds, pathogens, and plants can affect the structure of the indigenous food web, alter key abiotic variables such as groundwater, and cause reductions in local species diversity (Parker and Reichard 1998, Parker et al. 1999). At the same time, many introductions are unsuccessful or have no effect on the structure and function of indigenous communities (Williamson and Fitter 1996). A good deal of effort has been invested in trying to understand how the intrinsic properties of invasive species such as their reproductive rates, dietary requirements, and associations with humans (Bazzaz 1986) contribute to their success or failure. Although these studies have yielded a variety of useful case studies, they have mostly failed to establish general principles to guide managers in novel instances.
Orians (1986) argues that the shift of attention away from the receiving environment to the nature of the colonizing species has contributed to the lack of progress in deriving a more general understanding of invasions. He advocates an approach that combines information about the invading organisms with information about the environments into which they are being introduced and discusses the properties of the receiving environment in terms of disturbance, biotic interactions, and temporal scale. Similarly, Parker et al. (1999) ask to what extent the impact of an invasive species is dependent on the environment into which it is introduced.
Invasions usually come as ecological surprises. Although it may not be possible to make strong predictions about the likelihood of species invasions (May 1976), we can anticipate that invasions will be more likely in some places than in others. In particular, some habitats or habitat configurations are more susceptible to invasion than others. If we can identify the general properties of habitats that make them more or less vulnerable to invasion, we need to take these properties into consideration when designing and managing conservation areas. There are many characteristics of a region that will affect the likelihood of its being successfully invaded, including:
- geographic isolation and proximity to potential sources of invaders (MacArthur and Wilson 1967). It should be noted that habitat proximity may be natural or anthropogenic. For example, areas on transport routes or with migratory human populations will be more vulnerable to invasion;
- the number of similar habitats in other parts of the world;
- the productivity of the habitat and the nature of the biota currently living within it (Pacala and Roughgarden 1982);
- the nature and frequency of relevant disturbances (Elton 1958, Hobbs and Huenneke 1992);
- the ease with which organisms move through the habitat as determined by relevant features such as vegetation type, wind, or water flow rates;
- historical factors affecting the local species pool such as recent extinctions and previous invasions (Allen et al. 1999);
- the specific match of invading species to habitat; and
- habitat size and shape.
These and other factors, such as the rate of increase and minimum viable population
size of the invading species, will also determine how serious the consequences
of the invasion are and the likelihood that the invading species can be controlled.
An important correlate of the risk associated with a particular introduction is the rate at which individuals of the invading species can multiply and disperse. In this paper, I examine the question of whether habitat shape can affect rates of colonization. Previous studies of habitat shape that focused primarily on metapopulations and dispersal between habitat fragments (Hamazaki 1996, Dunham et al. 1997) often ignored the relevance of habitat shape at smaller and larger scales. Although Major et al. (1999) found significantly higher robin (Erithacus rubecula) densities in larger habitat fragments and different age structures in the robin populations of large and small fragments, the scale of their analysis was such that the results must necessarily be considered in terms of between-patch rather than within-patch dispersal. As I demonstrate using two different, spatially explicit simulation methods (a cellular automaton model and a reaction-diffusion model), these processes are affected by habitat shape. The models are used to generate hypotheses about the susceptibility of different habitats to invasion. In the future, it may be possible to use this kind of approach to calibrate the relative vulnerabilities of habitat fragments and reserves to invasions.
METHODS
The spread of organisms in hypothetical habitats of different shapes was modeled using two different methods: a stochastic cellular automaton and a reaction-diffusion model. The Matlab routines for each model are provided in Appendix 1. Both methods have been widely used to study invasions (Auld and Coote 1980, Okubo 1980, Roughgarden 1986, Hengeveld 1994, Tilman and Kareiva 1997). In both instances, I kept the design of the model as simple as possible to minimize the number of parameters that might affect the outcome of the modeling exercise. All comparisons were undertaken using identical parameters and habitats of equivalent sizes (i.e., the same number of square cells) that differed only in shape.
Hypothetical habitats
Each hypothetical habitat consisted of 1600 square cells (the habitat files are available in Appendix 2). Habitat shapes ranged from a straight line 1600 cells long to a 40 x 40 grid. The network habitats (Fig. 1) imitate streams of different complexities with 0, 10, and 20 nodes, respectively. Hypothetical streams of the same length and with the same number of nodes can be constructed at different levels of complexity, depending on how many ramifications their divergences have (i.e., how hierarchical they are); I constructed both a simple and a more complex model for each of the branching streams (10 and 20 nodes). Note that the complex 10-node and 20-node networks have slightly different hierarchical arrangements.
Fig. 1. Different hypothetical streamlike habitats compared in the modeling exercise (these drawings are not to scale). Each habitat consists of 1600 square cells; all parts of each network are the same width. Drawing (a) represents a linear habitat with no nodes; (b), a simple design with 10 nodes; (c), a complex design with 10 nodes; (d), a simple design with 20 nodes; and (e), a complex design with 20 nodes. Note that the complex 10-node habitat (c) has a different form from that of the complex 20-node habitat (e).
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The regularly shaped habitats imitate square or rectangular terrestrial areas measuring 10 x 160, 20 x 80, and 40 x 40 cells, respectively. I did not attempt to simulate the effects of increasing the number of corners, although corner cells will obviously have a lower connectivity than do perimeter cells.
Cellular automaton model
A thorough description of cellular automata in spatial ecology can be found in Tilman and Kareiva (1997). The model follows a simple, iterative form in which, at each time step, occupied cells can produce two offspring (these "propagules" can be thought of as subpopulations or migrating individuals). The offspring are assigned randomly to any one of the cells within a user-defined maximum dispersal distance of the parent cell (Fig. 2); this distance is set at three cell widths for all simulations. If an occupied cell (value = 1) is selected by the random function, no net change occurs; if an unoccupied cell is selected, its value changes from 0 to 1. The multiplication and dispersal processes are offset by a mortality level that ranges from 0 to 1. Mortality is imposed on each
cell by selecting a uniformly distributed random number between 0 and 1; at a mortality of 0.3, for example, a cell's value will change from 1 to 0 if the random number is less than or equal to 0.3.
Fig. 2. A flow chart showing the processes that occur within the cellular automaton model. The introduced population produces two propagules or offspring that disperse randomly within a set distance. If they land in an occupied cell, they die; if they land in an unoccupied cell, they survive. All occupied cells then undergo random mortality before the cycle repeats itself.
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Individual simulations were continued for 5000 iterations, by which time each simulation had reached equilibrium. Because the model is stochastic, it does not produce the same answer every time; although it approaches similar equilibrial densities, these may differ depending on random fluctuations. I used multiple replicates of each set of conditions to establish general trends. The program reports which cells are occupied at each iteration and the total number of cells occupied, which serves as a surrogate for metapopulation density. I initially explored the consequences of varying mortality rates using 20 replicates of each set of conditions. These simulations suggested that shape effects were more likely at higher mortality rates. I then ran 100 replicates in each hypothetical habitat type at a mortality of 0.3. These simulations included five stream habitat types (simple with 0 nodes and simple or complex with 10 or 20 nodes) and three grid habitat types (40 x 40, 20 x 80, and 10 x 160 cells, respectively).
For each hypothetical habitat, I considered the rate at which the simulation reached equilibrium and the total carrying capacity of each cell. The data were summarized using the total number of occupied cells at each time step. To determine an equilibrium value for cellular automaton simulations, I took the mean of the last 500 iterations. Time to equilibrium was defined as the time that elapsed before the total number of occupied cells first exceeded this mean value. A typical sequence of colonization through a real stream network in Wisconsin is shown in Figs. 3 and 4.
Fig. 3. Animated time series simulating colonization of a real stream habitat in Wisconsin by a hypothetical invasive species, as modeled using the cellular automaton model. The number in the upper left corner indicates the number of iterations at which the observation was made. The blue grid outlines habitat cells, and white shading indicates cell occupancy.
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Fig. 4. Plot showing the total size of the metapopulation of the real stream over time. As can be seen by watching Fig. 3, the species has reached all parts of the stream network by iteration 250; after this there is little net change in abundance, although the occupancies of different cells vary.
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Reaction-diffusion model
Roughgarden (1986) stated that the theory of the rate of spread of an invasion " ... is quite robust, has been empirically tested, and is about as reliable as theory gets." The theory to which she was referring originated with Fisher's (1937) population-genetic model for the spread of a favorable mutation and was first brought into a specifically ecological context by Skellam (1951). The central prediction, i.e., that the square root of the area occupied by the invading species grows linearly with time, follows from the coupling of a diffusion model with exponential population growth to produce a reaction-diffusion model. Although classical reaction-diffusion models have some shortcomings when applied to real data, because their assumption of a normal rather than a leptokurtic dispersal curve may result in the underestimation of the rate of spread of the invading population (Kot et al. 1996), this inaccuracy is irrelevant to the conclusions of this paper.
I used a simple form of reaction-diffusion model known as Fisher's equation. A full outline of mathematical and programming methods for diffusion models can be found in Press et al. (1992). Reaction-diffusion models have two parts: a diffusion term and a reaction term. Diffusion and reaction occur separately. The reaction term in this instance was a simple logistic population increase:
Nt+1 = Nt + rNt(1 - (Nt/K)) |
(1) |
where Nt is the population size at time t, r is the intrinsic rate of increase, and K is the carrying capacity (Tilman and Kareiva 1997). I did not include stochastic mortality in the population model. The population parameter r was set at 0.5, and K remained at 500 individuals per cell for all simulations.
The diffusion term used a multidimensional form of the diffusion equation (Press et al. 1992), which in one space dimension is
where D is the diffusion coefficient, t is time, u is the population size in a given cell, and x is the distance over which diffusion takes place. This equation can be differenced in various ways, the simplest of which (in a single dimension) is
(ujn+1 - ujn)/t = D(uj+1n - 2ujn + uj-1n)/(x)2 |
(3) |
where D is the diffusion coefficient, uj is the population size at time t, uj+1 and uj-1 indicate population sizes in neighbouring cells, n is the current iteration, t is the time step, and x is the distance over which diffusion takes place. I used this form of the equation because of its computational speed, which is important for simulations this large. Its main drawback is that it is only stable for parameter combinations such that
All the simulations used unit habitat cells and a time step of 1, but, because each cell had up to four neighbours, the value of the diffusion coefficient (D) was constrained to less than 0.25.
For each cell, the model considers how many neighbors it has and uses the appropriate diffusion term (i.e., for one, two, three, or four neighbors). Neighboring cells in the reaction-diffusion model are defined as those that share two corners; dispersal was permitted only between immediate neighbors. If run to equilibrium, the model will fill all available habitat to carrying capacity (Fig. 5).
Fig. 5. Animated time series simulating colonization of a 10 x 10-degree grid by a hypothetical invasive species, as modeled using a reaction-diffusion process. A population of 100 individuals is introduced at the centre of the grid, and logistic population increase, coupled with diffusion, occurs until the population reaches carrying capacity. The number in the upper left corner indicates the iteration number. The colors correspond to the values of individual cells along a gradient from red through blue and green to yellow and back to red; cells are reshaded at each iteration.
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The reaction-diffusion model was run in the same hypothetical habitats as the cellular automaton model. In each instance, 100 individuals were introduced into a single cell, and the simulation continued until all the cells in the habitat had reached carrying capacity. The change in total abundance over time was then plotted to give the rate at which the invasive metapopulation increased. Because this model has no stochastic terms, multiple simulations under the same conditions were not necessary. Because there was no mortality term, the final carrying capacity of each habitat was identical.
RESULTS
For both the cellular automaton and the reaction-diffusion models, different hypothetical habitats (Fig. 1) were colonized at different rates (Figs. 6 and 7). The populations in the two-dimensional grids were the first to reach equilibrium, followed by those in the branching networks; the linear system was the slowest to be invaded in each instance. The more hierarchically arranged networks were colonized more quickly than the networks with a less nested form but the same number of nodes.
Fig. 6. Rates of occupancy of different hypothetical habitats under the cellular automaton model (A) and the reaction-diffusion model (B). In each graph, streamN denotes a branching habitat with N nodes that can be either simple (s) or complex (c), and grdxxy denotes a square or rectangular habitat with sides of length x and y.
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Fig. 7. Bar charts showing the differences in time to equilibrium for different hypothetical habitats under each model. Chart A results from the cellular automaton model at a mortality of 0.3; B, from the reaction-diffusion model with a population parameter of 0.5 and a carrying capacity of 500. Please see Fig. 6 for an explanation of the captions on the x axes.
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The influence of habitat shape on population processes will clearly be less pronounced for populations that have shorter generation times, move farther or faster, or have higher fecundity. As one might expect, the lower mortality rates in the cellular automaton model resulted in more similar colonization rates (Fig. 8).
Fig. 8. Equilibrium abundances attained in hypothetical habitats by the cellular automaton model. The error bars indicate ± one standard deviation in the values of the last 500 iterations (each of these values is an average of 100 simulations). Please see Fig. 6 for an explanation of the captions on the x axis.
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Interestingly, the final abundance of metapopulations from the cellular automaton model (as measured by the mean and standard deviation of the last 500 iterations) was significantly different between habitats (Fig. 9). In general, habitats that were colonized more quickly achieved higher eventual abundances of invaders. A test for correlation between time to equilibrium and eventual abundance yielded Kendall's Tau = -0.571 with a significance of p = 0.048 (n = 8). This result was influenced by the unexpectedly high abundance of individuals in the linear system; with this observation removed from the analysis, the value of Kendall's Tau increased to -0.81 (p = 0.007, n = 7).
Fig. 9. Times to equilibrium, expressed as number of time steps, in the 10-node stream network under different mortality conditions. When the random mortality term is set to 0.05, the population occupies its habitat more quickly and reaches a higher carrying capacity than at a mortality of 0.3.
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DISCUSSION
The results suggest unequivocally that habitat shape is an important determinant of the rate at which an invading population can fill available space. Although these models ignore many real-world factors, they serve as a useful heuristic from which to generate hypotheses. Both modeling methods have been applied successfully in real-world situations (Okubo 1980, Levin 1992, Tilman and Kareiva 1997) and yielded the same intuitively obvious conclusion.
Habitats that have a more complex topology allow organisms to move through them at a faster rate, colonize unoccupied spaces more rapidly, and reach carrying capacity more quickly. Consequently, they are more likely than identical habitats of different topologies to be successfully invaded by an introduced species that is well suited to local conditions. The differences in the rates at which organisms can move within and between habitats may in turn create differences in the age structure of populations, depending on which life stages disperse more (Major et al. 1999), and the ultimate size of the population, depending
on the local disturbance regime and how quickly empty habitats are recolonized.
The variations in colonization rates among different hypothetical habitats are not a simple consequence of differing degrees of connectivity. Although connectivity is undoubtedly important, the difference in colonization rates between hypothetical networks with identical connectivities but different spatial arrangements (such as the simple and complex networks with 10 nodes) shows that the precise arrangement of cells relative to one another is also important for processes related to dispersal. A more important metric, but one that is harder
to quantify, is how branching or hierarchical the system is. Once an increasing population has split into two subpopulations at a node, it then increases at twice the previous rate. Faster colonization is facilitated by habitat configurations that split the population more frequently per unit area. Arrangements with more nodes are colonized more rapidly, but the position of these nodes relative to one another is also important. In real aquatic systems, this implies that more complex river networks will be more vulnerable to invasion than simpler ones;
in the terrestrial context, reserve arrangements or the manner in which habitats are fragmented will be a correlate of their susceptibility to invasion.
The current focus in the design of reserves and reserve networks is primarily on proximity and connectivity. The connection of protected areas by corridors (Simberloff et al. 1992, Collinge 2000, Mortberg and Wallentinus 2000) needs to be carefully thought out, particularly in regions where species invasions are common. Increasing the connectivity of reserves will not necessarily be beneficial to the cause of conservation. Habitats that are more complex may be more vulnerable to the spread of invaders, parasites, and pathogens. On the
positive side, they may also be more resilient to external perturbations because areas from which organisms are eliminated will be filled more quickly. The results of this study suggest that dispersal and population abundance will be influenced not only by the number of corridors but also by how hierarchical the resulting system is. In aquatic systems, where branching is more obvious, control of invasive species may be made more efficient by taking habitat complexity and topology into account; managers will need to move more rapidly to contain invasive species in habitats with complex topologies, where they will spread
more quickly and attain higher abundances, than in simpler catchments where the spread of the species is likely to be slower and the final impact smaller. Similarly, it may be easier to fragment complex habitats deliberately without loss of functionality by breaking key connections, thus limiting the spread of invaders through the system.
Adequate tests of these predictions are likely be difficult in real situations, where differences in the local environment between habitat fragments may obscure shape effects. The model world differs from the real world in many ways, most importantly in the simplifying assumptions that the models make about habitat homogeneity and constant rates of movement. Stream environments will probably be less homogeneous within highly branching networks than within relatively linear systems because of differences in the rates of change in stream size. In addition, the number of tributaries feeding into a stream is affected by superficial geomorphology, which also affects biotic communities. Large differences in tributary ramifications are unlikely to occur in highly similar habitats. Real-world populations typically show considerable variation in mortality rates; abiotic disturbances, human interventions, and biotic factors such as competition and predation are seldom constant in time or space. Similarly, animals typically disperse more at certain times of the year than at others. The best test systems will probably be those involving hardy, ubiquitous invaders that move along stream networks or riparian corridors; possible study species include crayfish and rabbits.
SPECULATION
As with other explicitly spatial phenomena, the relative differences generated by shape effects are likely to depend on scale. In real-world situations where there is a choice between different reserve designs, this kind of simulation approach may provide some valuable insights into the likely success of invasions and reintroductions. Dispersal and reproduction models that more closely imitate the spread of real animals in real habitats need to be developed and tested in real and hypothetical reserves of different designs. We can envisage that most reserve designs will be a compromise between risk and security, with smaller reserves being more vulnerable to disruption and larger reserves more robust. Conversely, control of invasive species will be more difficult in larger areas. Given more
information on the vulnerability of habitats to invasion, it may be possible at some point to calculate an optimal or minimum reserve size, shape, and connectivity for individual species to balance internally and externally driven processes. In the meantime, the most secure long-term strategy is probably to maintain reserves of different sizes and shapes (including some very isolated areas) and monitor them closely.
The shape-dependency of dispersal and reproduction processes over large areas and long time scales may also have important consequences for evolutionary processes such as speciation and extinction. Speciation is more likely in habitats where genetic isolation is more readily achieved. It stands to reason that the likelihood of an isolating event would be higher in longer, thinner habitats with fewer connections. Extinctions may also be more likely in less complex habitats; for example, the effects of global warming on cold-water fish species
may be more evident in high-altitude streams that have fewer branches.
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Acknowledgments:
I am grateful to Steve Carpenter and two anonymous referees for comments on an earlier draft of this manuscript. This work was funded by a David H. Smith postdoctoral fellowship from The Nature Conservancy. This is publication DHS2001-1 of the David H. Smith Conservation Research Fellowship Program.
APPENDIX 1
Append1.zip contains Matlab
version 5.3 routines for both of the models that are presented in this
paper. The files are in ASCII text format. Use the links below to view the files online.
The cellular automaton model Cellaut.m
contains matlab code for the cellular automaton model. It is currently set
to run 20 iterations, each of 5000 steps, under the given conditions.
The reaction-diffusion model This program consists of six
separate .m files. Diffuse.m is the master routine. Diffuse1.m to
Diffuse4.m are diffusion routines that are called by Diffuse.m for
cells with one to four neighbours. Reproduce.m contains the logistic
population increase term. This model provides a framework into which
any set of coordinates can be read. It requires that parameter
restrictions are observed (as outlined in the methods section) and
that the distance between cells is unity. Grids based on real
coordinates can easily be reformatted to give unit coordinate spacing;
simply subtract the minimum x and y coordinates and divide by the
inter-cell distance. If this is done in a spreadsheet, keeping the
key of matching coordinate pairs, the results can readily be matched
to the original coordinates and plotted in ‘real’
space. Note that dispersal can only occur between adjacent cells;
there is no long-distance dispersal to unconnected habitat patches.
In both cases you will need to change the working directory (from
which input files are read, and to which the results file is sent) by
altering the line ‘cd d:\myfolder’ to ‘cd
yourworkingdiskdrive:\yourfolderofchoice’. (e.g. change
‘d:\myfolder’ to d:\models\diffusion’). You may
also need to add your folder to Matlab’s directory using the
‘set path’ option in the Matlab command window.
cellaut.m
% Cellular automaton model - stochastic dispersal, reproduction and mortality
% Written by Graeme Cumming, November 2000
% set the working directory for your own machine
cd e:\cellaut2
% these commands read in x,y coords from file myfile.txt
fid = fopen ('stream0.txt','r');
[xcoord,ycoord] = textread ('stream0.txt','%f %f');
status = fclose (fid);
count = size (xcoord);
% initialise the variables for the program
cell = zeros (count);
cell2 = zeros (count);
cell (500) = 1; %starting value
dispdist = 3; %dispersal distance in m
inc = 2; %number of 'offspring' per iteration
mortality = 300; %death rate / 1000
area = ceil (3.141592654*(dispdist^2)); %max possible number of neighbours
neighbour = zeros (count,area);
reps = 5000;
%first we write a matrix containing identifiers of all cells in dispersal range
for x = 1:count
match = 1;
for i = 1:count
distance = sqrt(((xcoord(x)-xcoord(i))^2)+((ycoord(x)-ycoord(i))^2));
if distance <= dispdist
match = match + 1;
neighbour(x,1) = neighbour(x,1)+1;
neighbour(x,match) = i;
end;
end;
end;
%now begins the main loop of the program
for z = 1:100
cell = zeros (count);
cell2 = zeros (count);
cell (500) = 1;
results = zeros (10000,1);
for r = 1:reps
%disperse
for j = 1:count
if cell(j) == 1
cell2(j) = 1;
for m = 1:inc
rnum = rand * neighbour(j,1);
p = ceil(rnum);
if p==1
p = 2;
end;
plusone_id = neighbour(j,p);
cell2(plusone_id) = 1;
end;
end;
end;
cell = cell2;
cell2 = zeros (count);
for k = 1:count
if round(rand*1000) < mortality
cell(k) = 0;
end;
end;
%results
tally = sum(cell);
results(r,1) = tally;
if tally==0
break
end;
disp (r);
end;
%^ ends the main loop
if z == 1
final = results;
end;
if z > 1
final = cat(2,final,results);
end;
end;
save ('results1.txt','final','-ascii');
diffuse.m
% Program Diffuse
% Written by Graeme Cumming, October 2000
% Program aim is to provide a general reaction-diffusion modelling framework
% Coordinates of study system are read in from a text file 'coords'
% A matrix 'proxim' is then written that holds the number of adjacent cells
% and which cells are adjacent, for each cell
% Diffusion occurs in one of three ways depending on # adjacent cells
% 1 adjacent cell: simple exchange
% 2 adjacent cells: reaction-diffusion in one plane
% 3 adjacent cells: r-d between 3 peripheral and one central cells
% 4 adjacent cells: two dimensional r-d
% After diffusion is completed for each cell, populations increase
% Population increase is determined by a logistic growth equation
% start by setting the home directory
cd e:\diffusn\streams\
% these commands read in x,y coords from columns in file tlake1.txt
fid = fopen ('stream0.txt','r');
[xcoord,ycoord] = textread ('stream0.txt','%f %f');
status = fclose (fid);
count = size (xcoord);
% the coordinate data are now in two vectors, 'xcoord' and 'ycoord'
% 'count' holds the number of coordinates
global adjacent
adjacent = zeros (count,5);
% reserve space for number of and identifiers of adjacent cells
global current;
current = zeros (count);
% the vector 'current' holds current n for each cell at time t
global cplus1;
cplus1 = zeros (count);
% 'cplus1' holds n for each cell at time step t+1
global D;
D = 0.1;
% D is the diffusion coefficient
% for the current diffuse2 it must be <= .5
global cellwidth;
cellwidth = 1;
% the real-world width of the cells, which are presumed to be squares
% now to calculate the proximity key
for j = 1:count
adjacent(j,1) = 0;
for i = 1:count
distance = sqrt(((xcoord(i)-xcoord(j))^2)+((ycoord(i)-ycoord(j))^2));
if distance == cellwidth
adjacent(j,1) = adjacent(j,1) + 1;
celltofill = adjacent(j,1) + 1;
adjacent(j,celltofill) = i;
end;
end;
end;
% the 5 x count matrix has number of neighbours and up to 4 cell identifiers
% for each cell in the analysis
% seed the 500th cell with 100 individuals
current(500) = 100;
% next, the main diffusion loop
iterations = 3000;
step = 1;
%results = current;
num = 1;
results = zeros (count);
for k = 1:iterations
% number of diffusion iterations
% the following calls different diffusion routines
for j = 1:count
switch (adjacent(j,1))
case (1)
diffuse1 (j);
case (2)
diffuse2 (j);
case (3)
diffuse3 (j);
case (4)
diffuse4 (j);
otherwise
disp ('ERROR - not finding neighbours');
end;
reproduce (j);
end;
current = cplus1;
% save the results every n steps
if k/step == floor(k/step)
% results = cat(2,results,cplus1);
results(num) = sum(cplus1);
num = num+1;
end;
disp (k)
if k/3000 == floor(k/3000)
save ('results1.txt','results','-ascii');
end;
end;
diffuse1.m to diffuse4.m diffuse1.m
function [diff1] = diffuse1(j)
% diffusion into the end-cell
global cplus1;
global adjacent;
global current;
global D;
global cellwidth;
neighbour = adjacent(j,2);
unj = current(neighbour);
% unj is the current value of the cell next to the end-point
endcell = current(j);
if endcell <= unj
cplus1(j) = endcell + (unj-endcell)/2;
end;
if endcell > unj
cplus1(j) = endcell - (endcell-unj)/2;
end;
diffuse2.m function [diff2] = diffuse2(j)
global cplus1;
global current;
global adjacent;
global D;
global cellwidth;
unj = current(j);
currnt1 = adjacent(j,2);
unjplus1 = current (currnt1);
currnt2 = adjacent (j,3);
unjminus1 = current (currnt2);
RHS = (((unjplus1 - (2*unj) + unjminus1))/(cellwidth^2))*D;
% cellwidth = delta x
cplus1(j) = RHS + unj;
diffuse3.m
function [diff3] = diffuse3(j)
global cplus1;
global current;
global adjacent;
global D;
global cellwidth;
unj = current(j);
c1 = adjacent(j,2);
c2 = adjacent(j,3);
c3 = adjacent(j,4);
% c1-c3 are the identifiers of the 3 neighbouring cells
currnt1 = current(c1);
currnt2 = current(c2);
currnt3 = current(c3);
% currnt1-currnt3 are the values of the 3 neighbouring cells
RHS = (((currnt1 - (3*unj) + currnt2 + currnt3))/(cellwidth^2))*D;
% cellwidth = delta x = 1; delta t is 1
% the flow is now three-way
cplus1(j) = RHS + unj;
diffuse4.m
function [diff4] = diffuse4(j)
global cplus1;
global current;
global adjacent;
global D;
global cellwidth;
unj = current(j);
c1 = adjacent(j,2);
c2 = adjacent(j,3);
c3 = adjacent(j,4);
c4 = adjacent(j,5);
% c1-c4 are the identifiers of the 4 neighbouring cells
currnt1 = current(c1);
currnt2 = current(c2);
currnt3 = current(c3);
currnt4 = current(c4);
% currnt1-currnt4 are the values of the 4 neighbouring cells
RHS = (((currnt1 - (4*unj) + currnt2 + currnt3 + currnt4))/(cellwidth^2))*D;
% cellwidth = delta x = 1; delta t is also 1
% the flow is now four-way
cplus1(j) = RHS + unj;
reproduce.m
function [shithappens] = reproduce (j)
% exponential population increase
global cplus1;
N = cplus1(j);
r = .5;
K = 500;
babies = r*N*(1-(N/K));
%fate = rand;
% fate is a random number between 0 and 1
% corpses = ((fate/1)*N)/2;
% a random proportion of the population, not exceeding 50%, dies at every iteration
corpses = 0;
cplus1(j) = cplus1(j) + babies - corpses;
APPENDIX 2
Append2.zip contains the hypothetical habitat
files used in the analysis. Each pair of (x,y) coordinates specifies
the centre of a habitat cell. The files are in ASCII text format; PC
users can open them through Excel or view them directly in Notepad.
grd10x160
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5 15
6 15
7 15
8 15
9 15
10 15
11 15
12 15
13 15
14 15
15 15
16 15
17 15
18 15
19 15
20 15
1 16
2 16
3 16
4 16
5 16
6 16
7 16
8 16
9 16
10 16
11 16
12 16
13 16
14 16
15 16
16 16
17 16
18 16
19 16
20 16
1 17
2 17
3 17
4 17
5 17
6 17
7 17
8 17
9 17
10 17
11 17
12 17
13 17
14 17
15 17
16 17
17 17
18 17
19 17
20 17
1 18
2 18
3 18
4 18
5 18
6 18
7 18
8 18
9 18
10 18
11 18
12 18
13 18
14 18
15 18
16 18
17 18
18 18
19 18
20 18
1 19
2 19
3 19
4 19
5 19
6 19
7 19
8 19
9 19
10 19
11 19
12 19
13 19
14 19
15 19
16 19
17 19
18 19
19 19
20 19
1 20
2 20
3 20
4 20
5 20
6 20
7 20
8 20
9 20
10 20
11 20
12 20
13 20
14 20
15 20
16 20
17 20
18 20
19 20
20 20
1 21
2 21
3 21
4 21
5 21
6 21
7 21
8 21
9 21
10 21
11 21
12 21
13 21
14 21
15 21
16 21
17 21
18 21
19 21
20 21
1 22
2 22
3 22
4 22
5 22
6 22
7 22
8 22
9 22
10 22
11 22
12 22
13 22
14 22
15 22
16 22
17 22
18 22
19 22
20 22
1 23
2 23
3 23
4 23
5 23
6 23
7 23
8 23
9 23
10 23
11 23
12 23
13 23
14 23
15 23
16 23
17 23
18 23
19 23
20 23
1 24
2 24
3 24
4 24
5 24
6 24
7 24
8 24
9 24
10 24
11 24
12 24
13 24
14 24
15 24
16 24
17 24
18 24
19 24
20 24
1 25
2 25
3 25
4 25
5 25
6 25
7 25
8 25
9 25
10 25
11 25
12 25
13 25
14 25
15 25
16 25
17 25
18 25
19 25
20 25
1 26
2 26
3 26
4 26
5 26
6 26
7 26
8 26
9 26
10 26
11 26
12 26
13 26
14 26
15 26
16 26
17 26
18 26
19 26
20 26
1 27
2 27
3 27
4 27
5 27
6 27
7 27
8 27
9 27
10 27
11 27
12 27
13 27
14 27
15 27
16 27
17 27
18 27
19 27
20 27
1 28
2 28
3 28
4 28
5 28
6 28
7 28
8 28
9 28
10 28
11 28
12 28
13 28
14 28
15 28
16 28
17 28
18 28
19 28
20 28
1 29
2 29
3 29
4 29
5 29
6 29
7 29
8 29
9 29
10 29
11 29
12 29
13 29
14 29
15 29
16 29
17 29
18 29
19 29
20 29
1 30
2 30
3 30
4 30
5 30
6 30
7 30
8 30
9 30
10 30
11 30
12 30
13 30
14 30
15 30
16 30
17 30
18 30
19 30
20 30
1 31
2 31
3 31
4 31
5 31
6 31
7 31
8 31
9 31
10 31
11 31
12 31
13 31
14 31
15 31
16 31
17 31
18 31
19 31
20 31
1 32
2 32
3 32
4 32
5 32
6 32
7 32
8 32
9 32
10 32
11 32
12 32
13 32
14 32
15 32
16 32
17 32
18 32
19 32
20 32
1 33
2 33
3 33
4 33
5 33
6 33
7 33
8 33
9 33
10 33
11 33
12 33
13 33
14 33
15 33
16 33
17 33
18 33
19 33
20 33
1 34
2 34
3 34
4 34
5 34
6 34
7 34
8 34
9 34
10 34
11 34
12 34
13 34
14 34
15 34
16 34
17 34
18 34
19 34
20 34
1 35
2 35
3 35
4 35
5 35
6 35
7 35
8 35
9 35
10 35
11 35
12 35
13 35
14 35
15 35
16 35
17 35
18 35
19 35
20 35
1 36
2 36
3 36
4 36
5 36
6 36
7 36
8 36
9 36
10 36
11 36
12 36
13 36
14 36
15 36
16 36
17 36
18 36
19 36
20 36
1 37
2 37
3 37
4 37
5 37
6 37
7 37
8 37
9 37
10 37
11 37
12 37
13 37
14 37
15 37
16 37
17 37
18 37
19 37
20 37
1 38
2 38
3 38
4 38
5 38
6 38
7 38
8 38
9 38
10 38
11 38
12 38
13 38
14 38
15 38
16 38
17 38
18 38
19 38
20 38
1 39
2 39
3 39
4 39
5 39
6 39
7 39
8 39
9 39
10 39
11 39
12 39
13 39
14 39
15 39
16 39
17 39
18 39
19 39
20 39
1 40
2 40
3 40
4 40
5 40
6 40
7 40
8 40
9 40
10 40
11 40
12 40
13 40
14 40
15 40
16 40
17 40
18 40
19 40
20 40
1 41
2 41
3 41
4 41
5 41
6 41
7 41
8 41
9 41
10 41
11 41
12 41
13 41
14 41
15 41
16 41
17 41
18 41
19 41
20 41
1 42
2 42
3 42
4 42
5 42
6 42
7 42
8 42
9 42
10 42
11 42
12 42
13 42
14 42
15 42
16 42
17 42
18 42
19 42
20 42
1 43
2 43
3 43
4 43
5 43
6 43
7 43
8 43
9 43
10 43
11 43
12 43
13 43
14 43
15 43
16 43
17 43
18 43
19 43
20 43
1 44
2 44
3 44
4 44
5 44
6 44
7 44
8 44
9 44
10 44
11 44
12 44
13 44
14 44
15 44
16 44
17 44
18 44
19 44
20 44
1 45
2 45
3 45
4 45
5 45
6 45
7 45
8 45
9 45
10 45
11 45
12 45
13 45
14 45
15 45
16 45
17 45
18 45
19 45
20 45
1 46
2 46
3 46
4 46
5 46
6 46
7 46
8 46
9 46
10 46
11 46
12 46
13 46
14 46
15 46
16 46
17 46
18 46
19 46
20 46
1 47
2 47
3 47
4 47
5 47
6 47
7 47
8 47
9 47
10 47
11 47
12 47
13 47
14 47
15 47
16 47
17 47
18 47
19 47
20 47
1 48
2 48
3 48
4 48
5 48
6 48
7 48
8 48
9 48
10 48
11 48
12 48
13 48
14 48
15 48
16 48
17 48
18 48
19 48
20 48
1 49
2 49
3 49
4 49
5 49
6 49
7 49
8 49
9 49
10 49
11 49
12 49
13 49
14 49
15 49
16 49
17 49
18 49
19 49
20 49
1 50
2 50
3 50
4 50
5 50
6 50
7 50
8 50
9 50
10 50
11 50
12 50
13 50
14 50
15 50
16 50
17 50
18 50
19 50
20 50
1 51
2 51
3 51
4 51
5 51
6 51
7 51
8 51
9 51
10 51
11 51
12 51
13 51
14 51
15 51
16 51
17 51
18 51
19 51
20 51
1 52
2 52
3 52
4 52
5 52
6 52
7 52
8 52
9 52
10 52
11 52
12 52
13 52
14 52
15 52
16 52
17 52
18 52
19 52
20 52
1 53
2 53
3 53
4 53
5 53
6 53
7 53
8 53
9 53
10 53
11 53
12 53
13 53
14 53
15 53
16 53
17 53
18 53
19 53
20 53
1 54
2 54
3 54
4 54
5 54
6 54
7 54
8 54
9 54
10 54
11 54
12 54
13 54
14 54
15 54
16 54
17 54
18 54
19 54
20 54
1 55
2 55
3 55
4 55
5 55
6 55
7 55
8 55
9 55
10 55
11 55
12 55
13 55
14 55
15 55
16 55
17 55
18 55
19 55
20 55
1 56
2 56
3 56
4 56
5 56
6 56
7 56
8 56
9 56
10 56
11 56
12 56
13 56
14 56
15 56
16 56
17 56
18 56
19 56
20 56
1 57
2 57
3 57
4 57
5 57
6 57
7 57
8 57
9 57
10 57
11 57
12 57
13 57
14 57
15 57
16 57
17 57
18 57
19 57
20 57
1 58
2 58
3 58
4 58
5 58
6 58
7 58
8 58
9 58
10 58
11 58
12 58
13 58
14 58
15 58
16 58
17 58
18 58
19 58
20 58
1 59
2 59
3 59
4 59
5 59
6 59
7 59
8 59
9 59
10 59
11 59
12 59
13 59
14 59
15 59
16 59
17 59
18 59
19 59
20 59
1 60
2 60
3 60
4 60
5 60
6 60
7 60
8 60
9 60
10 60
11 60
12 60
13 60
14 60
15 60
16 60
17 60
18 60
19 60
20 60
1 61
2 61
3 61
4 61
5 61
6 61
7 61
8 61
9 61
10 61
11 61
12 61
13 61
14 61
15 61
16 61
17 61
18 61
19 61
20 61
1 62
2 62
3 62
4 62
5 62
6 62
7 62
8 62
9 62
10 62
11 62
12 62
13 62
14 62
15 62
16 62
17 62
18 62
19 62
20 62
1 63
2 63
3 63
4 63
5 63
6 63
7 63
8 63
9 63
10 63
11 63
12 63
13 63
14 63
15 63
16 63
17 63
18 63
19 63
20 63
1 64
2 64
3 64
4 64
5 64
6 64
7 64
8 64
9 64
10 64
11 64
12 64
13 64
14 64
15 64
16 64
17 64
18 64
19 64
20 64
1 65
2 65
3 65
4 65
5 65
6 65
7 65
8 65
9 65
10 65
11 65
12 65
13 65
14 65
15 65
16 65
17 65
18 65
19 65
20 65
1 66
2 66
3 66
4 66
5 66
6 66
7 66
8 66
9 66
10 66
11 66
12 66
13 66
14 66
15 66
16 66
17 66
18 66
19 66
20 66
1 67
2 67
3 67
4 67
5 67
6 67
7 67
8 67
9 67
10 67
11 67
12 67
13 67
14 67
15 67
16 67
17 67
18 67
19 67
20 67
1 68
2 68
3 68
4 68
5 68
6 68
7 68
8 68
9 68
10 68
11 68
12 68
13 68
14 68
15 68
16 68
17 68
18 68
19 68
20 68
1 69
2 69
3 69
4 69
5 69
6 69
7 69
8 69
9 69
10 69
11 69
12 69
13 69
14 69
15 69
16 69
17 69
18 69
19 69
20 69
1 70
2 70
3 70
4 70
5 70
6 70
7 70
8 70
9 70
10 70
11 70
12 70
13 70
14 70
15 70
16 70
17 70
18 70
19 70
20 70
1 71
2 71
3 71
4 71
5 71
6 71
7 71
8 71
9 71
10 71
11 71
12 71
13 71
14 71
15 71
16 71
17 71
18 71
19 71
20 71
1 72
2 72
3 72
4 72
5 72
6 72
7 72
8 72
9 72
10 72
11 72
12 72
13 72
14 72
15 72
16 72
17 72
18 72
19 72
20 72
1 73
2 73
3 73
4 73
5 73
6 73
7 73
8 73
9 73
10 73
11 73
12 73
13 73
14 73
15 73
16 73
17 73
18 73
19 73
20 73
1 74
2 74
3 74
4 74
5 74
6 74
7 74
8 74
9 74
10 74
11 74
12 74
13 74
14 74
15 74
16 74
17 74
18 74
19 74
20 74
1 75
2 75
3 75
4 75
5 75
6 75
7 75
8 75
9 75
10 75
11 75
12 75
13 75
14 75
15 75
16 75
17 75
18 75
19 75
20 75
1 76
2 76
3 76
4 76
5 76
6 76
7 76
8 76
9 76
10 76
11 76
12 76
13 76
14 76
15 76
16 76
17 76
18 76
19 76
20 76
1 77
2 77
3 77
4 77
5 77
6 77
7 77
8 77
9 77
10 77
11 77
12 77
13 77
14 77
15 77
16 77
17 77
18 77
19 77
20 77
1 78
2 78
3 78
4 78
5 78
6 78
7 78
8 78
9 78
10 78
11 78
12 78
13 78
14 78
15 78
16 78
17 78
18 78
19 78
20 78
1 79
2 79
3 79
4 79
5 79
6 79
7 79
8 79
9 79
10 79
11 79
12 79
13 79
14 79
15 79
16 79
17 79
18 79
19 79
20 79
1 80
2 80
3 80
4 80
5 80
6 80
7 80
8 80
9 80
10 80
11 80
12 80
13 80
14 80
15 80
16 80
17 80
18 80
19 80
20 80
grd40x40
1 1
2 1
3 1
4 1
5 1
6 1
7 1
8 1
9 1
10 1
11 1
12 1
13 1
14 1
15 1
16 1
17 1
18 1
19 1
20 1
21 1
22 1
23 1
24 1
25 1
26 1
27 1
28 1
29 1
30 1
31 1
32 1
33 1
34 1
35 1
36 1
37 1
38 1
39 1
40 1
1 2
2 2
3 2
4 2
5 2
6 2
7 2
8 2
9 2
10 2
11 2
12 2
13 2
14 2
15 2
16 2
17 2
18 2
19 2
20 2
21 2
22 2
23 2
24 2
25 2
26 2
27 2
28 2
29 2
30 2
31 2
32 2
33 2
34 2
35 2
36 2
37 2
38 2
39 2
40 2
1 3
2 3
3 3
4 3
5 3
6 3
7 3
8 3
9 3
10 3
11 3
12 3
13 3
14 3
15 3
16 3
17 3
18 3
19 3
20 3
21 3
22 3
23 3
24 3
25 3
26 3
27 3
28 3
29 3
30 3
31 3
32 3
33 3
34 3
35 3
36 3
37 3
38 3
39 3
40 3
1 4
2 4
3 4
4 4
5 4
6 4
7 4
8 4
9 4
10 4
11 4
12 4
13 4
14 4
15 4
16 4
17 4
18 4
19 4
20 4
21 4
22 4
23 4
24 4
25 4
26 4
27 4
28 4
29 4
30 4
31 4
32 4
33 4
34 4
35 4
36 4
37 4
38 4
39 4
40 4
1 5
2 5
3 5
4 5
5 5
6 5
7 5
8 5
9 5
10 5
11 5
12 5
13 5
14 5
15 5
16 5
17 5
18 5
19 5
20 5
21 5
22 5
23 5
24 5
25 5
26 5
27 5
28 5
29 5
30 5
31 5
32 5
33 5
34 5
35 5
36 5
37 5
38 5
39 5
40 5
1 6
2 6
3 6
4 6
5 6
6 6
7 6
8 6
9 6
10 6
11 6
12 6
13 6
14 6
15 6
16 6
17 6
18 6
19 6
20 6
21 6
22 6
23 6
24 6
25 6
26 6
27 6
28 6
29 6
30 6
31 6
32 6
33 6
34 6
35 6
36 6
37 6
38 6
39 6
40 6
1 7
2 7
3 7
4 7
5 7
6 7
7 7
8 7
9 7
10 7
11 7
12 7
13 7
14 7
15 7
16 7
17 7
18 7
19 7
20 7
21 7
22 7
23 7
24 7
25 7
26 7
27 7
28 7
29 7
30 7
31 7
32 7
33 7
34 7
35 7
36 7
37 7
38 7
39 7
40 7
1 8
2 8
3 8
4 8
5 8
6 8
7 8
8 8
9 8
10 8
11 8
12 8
13 8
14 8
15 8
16 8
17 8
18 8
19 8
20 8
21 8
22 8
23 8
24 8
25 8
26 8
27 8
28 8
29 8
30 8
31 8
32 8
33 8
34 8
35 8
36 8
37 8
38 8
39 8
40 8
1 9
2 9
3 9
4 9
5 9
6 9
7 9
8 9
9 9
10 9
11 9
12 9
13 9
14 9
15 9
16 9
17 9
18 9
19 9
20 9
21 9
22 9
23 9
24 9
25 9
26 9
27 9
28 9
29 9
30 9
31 9
32 9
33 9
34 9
35 9
36 9
37 9
38 9
39 9
40 9
1 10
2 10
3 10
4 10
5 10
6 10
7 10
8 10
9 10
10 10
11 10
12 10
13 10
14 10
15 10
16 10
17 10
18 10
19 10
20 10
21 10
22 10
23 10
24 10
25 10
26 10
27 10
28 10
29 10
30 10
31 10
32 10
33 10
34 10
35 10
36 10
37 10
38 10
39 10
40 10
1 11
2 11
3 11
4 11
5 11
6 11
7 11
8 11
9 11
10 11
11 11
12 11
13 11
14 11
15 11
16 11
17 11
18 11
19 11
20 11
21 11
22 11
23 11
24 11
25 11
26 11
27 11
28 11
29 11
30 11
31 11
32 11
33 11
34 11
35 11
36 11
37 11
38 11
39 11
40 11
1 12
2 12
3 12
4 12
5 12
6 12
7 12
8 12
9 12
10 12
11 12
12 12
13 12
14 12
15 12
16 12
17 12
18 12
19 12
20 12
21 12
22 12
23 12
24 12
25 12
26 12
27 12
28 12
29 12
30 12
31 12
32 12
33 12
34 12
35 12
36 12
37 12
38 12
39 12
40 12
1 13
2 13
3 13
4 13
5 13
6 13
7 13
8 13
9 13
10 13
11 13
12 13
13 13
14 13
15 13
16 13
17 13
18 13
19 13
20 13
21 13
22 13
23 13
24 13
25 13
26 13
27 13
28 13
29 13
30 13
31 13
32 13
33 13
34 13
35 13
36 13
37 13
38 13
39 13
40 13
1 14
2 14
3 14
4 14
5 14
6 14
7 14
8 14
9 14
10 14
11 14
12 14
13 14
14 14
15 14
16 14
17 14
18 14
19 14
20 14
21 14
22 14
23 14
24 14
25 14
26 14
27 14
28 14
29 14
30 14
31 14
32 14
33 14
34 14
35 14
36 14
37 14
38 14
39 14
40 14
1 15
2 15
3 15
4 15
5 15
6 15
7 15
8 15
9 15
10 15
11 15
12 15
13 15
14 15
15 15
16 15
17 15
18 15
19 15
20 15
21 15
22 15
23 15
24 15
25 15
26 15
27 15
28 15
29 15
30 15
31 15
32 15
33 15
34 15
35 15
36 15
37 15
38 15
39 15
40 15
1 16
2 16
3 16
4 16
5 16
6 16
7 16
8 16
9 16
10 16
11 16
12 16
13 16
14 16
15 16
16 16
17 16
18 16
19 16
20 16
21 16
22 16
23 16
24 16
25 16
26 16
27 16
28 16
29 16
30 16
31 16
32 16
33 16
34 16
35 16
36 16
37 16
38 16
39 16
40 16
1 17
2 17
3 17
4 17
5 17
6 17
7 17
8 17
9 17
10 17
11 17
12 17
13 17
14 17
15 17
16 17
17 17
18 17
19 17
20 17
21 17
22 17
23 17
24 17
25 17
26 17
27 17
28 17
29 17
30 17
31 17
32 17
33 17
34 17
35 17
36 17
37 17
38 17
39 17
40 17
1 18
2 18
3 18
4 18
5 18
6 18
7 18
8 18
9 18
10 18
11 18
12 18
13 18
14 18
15 18
16 18
17 18
18 18
19 18
20 18
21 18
22 18
23 18
24 18
25 18
26 18
27 18
28 18
29 18
30 18
31 18
32 18
33 18
34 18
35 18
36 18
37 18
38 18
39 18
40 18
1 19
2 19
3 19
4 19
5 19
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100 699
100 700
100 701
100 702
100 703
100 704
100 705
100 706
100 707
100 708
100 709
100 710
100 711
100 712
100 713
100 714
100 715
100 716
100 717
100 718
100 719
20 720
21 720
22 720
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24 720
25 720
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27 720
28 720
29 720
30 720
31 720
32 720
33 720
34 720
35 720
36 720
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44 720
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61 720
62 720
63 720
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65 720
66 720
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68 720
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72 720
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100 750
100 751
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100 768
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100 773
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100 777
100 778
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100 780
100 781
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100 790
100 791
100 792
100 793
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100 795
100 796
100 797
100 798
100 799
100 800
stream20
20 1
21 1
22 1
23 1
24 1
25 1
26 1
27 1
28 1
29 1
30 1
31 1
32 1
33 1
34 1
35 1
36 1
37 1
38 1
39 1
40 1
41 1
42 1
43 1
44 1
45 1
46 1
47 1
48 1
49 1
50 1
51 1
52 1
53 1
54 1
55 1
56 1
57 1
58 1
59 1
60 1
40 2
40 3
40 4
40 5
40 6
40 7
40 8
40 9
40 10
40 11
40 12
40 13
40 14
40 15
40 16
40 17
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40 24
40 25
40 26
40 27
40 28
40 29
40 30
40 31
40 32
40 33
40 34
40 35
40 36
40 37
40 38
40 39
1 40
2 40
3 40
4 40
5 40
6 40
7 40
8 40
9 40
10 40
11 40
12 40
13 40
14 40
15 40
16 40
17 40
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40 63
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40 65
40 66
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40 81
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40 104
40 105
40 106
40 107
40 108
40 109
40 110
40 111
40 112
40 113
40 114
40 115
40 116
40 117
40 118
40 119
1 120
2 120
3 120
4 120
5 120
6 120
7 120
8 120
9 120
10 120
11 120
12 120
13 120
14 120
15 120
16 120
17 120
18 120
19 120
20 120
21 120
22 120
23 120
24 120
25 120
26 120
27 120
28 120
29 120
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31 120
32 120
33 120
34 120
35 120
36 120
37 120
38 120
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40 160
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42 160
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48 160
49 160
50 160
51 160
52 160
53 160
54 160
55 160
56 160
57 160
58 160
59 160
60 160
61 160
62 160
63 160
64 160
65 160
66 160
67 160
68 160
69 160
70 160
71 160
72 160
73 160
74 160
75 160
76 160
77 160
78 160
79 160
80 160
40 161
40 162
40 163
40 164
40 165
40 166
40 167
40 168
40 169
40 170
40 171
40 172
40 173
40 174
40 175
40 176
40 177
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40 179
40 180
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40 182
40 183
40 184
40 185
40 186
40 187
40 188
40 189
40 190
40 191
40 192
40 193
40 194
40 195
40 196
40 197
40 198
40 199
1 200
2 200
3 200
4 200
5 200
6 200
7 200
8 200
9 200
10 200
11 200
12 200
13 200
14 200
15 200
16 200
17 200
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72 240
73 240
74 240
75 240
76 240
77 240
78 240
79 240
80 240
40 241
40 242
40 243
40 244
40 245
40 246
40 247
40 248
40 249
40 250
40 251
40 252
40 253
40 254
40 255
40 256
40 257
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40 259
40 260
40 261
40 262
40 263
40 264
40 265
40 266
40 267
40 268
40 269
40 270
40 271
40 272
40 273
40 274
40 275
40 276
40 277
40 278
40 279
1 280
2 280
3 280
4 280
5 280
6 280
7 280
8 280
9 280
10 280
11 280
12 280
13 280
14 280
15 280
16 280
17 280
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48 320
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51 320
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55 320
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58 320
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60 320
61 320
62 320
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64 320
65 320
66 320
67 320
68 320
69 320
70 320
71 320
72 320
73 320
74 320
75 320
76 320
77 320
78 320
79 320
80 320
40 321
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40 328
40 329
40 330
40 331
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40 335
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40 344
40 345
40 346
40 347
40 348
40 349
40 350
40 351
40 352
40 353
40 354
40 355
40 356
40 357
40 358
40 359
1 360
2 360
3 360
4 360
5 360
6 360
7 360
8 360
9 360
10 360
11 360
12 360
13 360
14 360
15 360
16 360
17 360
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25 360
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31 360
32 360
33 360
34 360
35 360
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38 360
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40 364
40 365
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48 400
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51 400
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55 400
56 400
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61 400
62 400
63 400
64 400
65 400
66 400
67 400
68 400
69 400
70 400
71 400
72 400
73 400
74 400
75 400
76 400
77 400
78 400
79 400
80 400
40 401
40 402
40 403
40 404
40 405
40 406
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40 408
40 409
40 410
40 411
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40 414
40 415
40 416
40 417
40 418
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40 420
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40 424
40 425
40 426
40 427
40 428
40 429
40 430
40 431
40 432
40 433
40 434
40 435
40 436
40 437
40 438
40 439
1 440
2 440
3 440
4 440
5 440
6 440
7 440
8 440
9 440
10 440
11 440
12 440
13 440
14 440
15 440
16 440
17 440
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25 440
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33 440
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35 440
36 440
37 440
38 440
39 440
40 440
40 441
40 442
40 443
40 444
40 445
40 446
40 447
40 448
40 449
40 450
40 451
40 452
40 453
40 454
40 455
40 456
40 457
40 458
40 459
40 460
40 461
40 462
40 463
40 464
40 465
40 466
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40 468
40 469
40 470
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40 480
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42 480
43 480
44 480
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55 480
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66 480
67 480
68 480
69 480
70 480
71 480
72 480
73 480
74 480
75 480
76 480
77 480
78 480
79 480
80 480
40 481
40 482
40 483
40 484
40 485
40 486
40 487
40 488
40 489
40 490
40 491
40 492
40 493
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40 500
40 501
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40 503
40 504
40 505
40 506
40 507
40 508
40 509
40 510
40 511
40 512
40 513
40 514
40 515
40 516
40 517
40 518
40 519
1 520
2 520
3 520
4 520
5 520
6 520
7 520
8 520
9 520
10 520
11 520
12 520
13 520
14 520
15 520
16 520
17 520
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25 520
26 520
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29 520
30 520
31 520
32 520
33 520
34 520
35 520
36 520
37 520
38 520
39 520
40 520
40 521
40 522
40 523
40 524
40 525
40 526
40 527
40 528
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40 530
40 531
40 532
40 533
40 534
40 535
40 536
40 537
40 538
40 539
40 540
40 541
40 542
40 543
40 544
40 545
40 546
40 547
40 548
40 549
40 550
40 551
40 552
40 553
40 554
40 555
40 556
40 557
40 558
40 559
40 560
41 560
42 560
43 560
44 560
45 560
46 560
47 560
48 560
49 560
50 560
51 560
52 560
53 560
54 560
55 560
56 560
57 560
58 560
59 560
60 560
61 560
62 560
63 560
64 560
65 560
66 560
67 560
68 560
69 560
70 560
71 560
72 560
73 560
74 560
75 560
76 560
77 560
78 560
79 560
80 560
40 561
40 562
40 563
40 564
40 565
40 566
40 567
40 568
40 569
40 570
40 571
40 572
40 573
40 574
40 575
40 576
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40 578
40 579
40 580
40 581
40 582
40 583
40 584
40 585
40 586
40 587
40 588
40 589
40 590
40 591
40 592
40 593
40 594
40 595
40 596
40 597
40 598
40 599
1 600
2 600
3 600
4 600
5 600
6 600
7 600
8 600
9 600
10 600
11 600
12 600
13 600
14 600
15 600
16 600
17 600
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32 600
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35 600
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40 603
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40 609
40 610
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40 612
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40 614
40 615
40 616
40 617
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40 619
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65 640
66 640
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68 640
69 640
70 640
71 640
72 640
73 640
74 640
75 640
76 640
77 640
78 640
79 640
80 640
40 641
40 642
40 643
40 644
40 645
40 646
40 647
40 648
40 649
40 650
40 651
40 652
40 653
40 654
40 655
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40 660
40 661
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40 667
40 668
40 669
40 670
40 671
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40 673
40 674
40 675
40 676
40 677
40 678
40 679
1 680
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3 680
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5 680
6 680
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11 680
12 680
13 680
14 680
15 680
16 680
17 680
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40 681
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40 700
40 701
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40 708
40 709
40 710
40 711
40 712
40 713
40 714
40 715
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40 717
40 718
40 719
40 720
41 720
42 720
43 720
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46 720
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76 720
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79 720
80 720
40 721
40 722
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40 724
40 725
40 726
40 727
40 728
40 729
40 730
40 731
40 732
40 733
40 734
40 735
40 736
40 737
40 738
40 739
40 740
40 741
40 742
40 743
40 744
40 745
40 746
40 747
40 748
40 749
40 750
40 751
40 752
40 753
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40 755
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40 758
40 759
1 760
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3 760
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40 780
40 781
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40 788
40 789
40 790
40 791
40 792
40 793
40 794
40 795
40 796
40 797
40 798
40 799
40 800
0 40
0 120
0 200
0 280
0 360
0 440
0 520
0 600
0 680
0 760
stream10c
0 7
0 8
0 9
0 10
0 11
0 12
0 13
0 14
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0 16
0 17
0 18
0 19
0 20
0 21
0 22
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1 42
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3 42
4 42
5 42
6 42
7 42
8 42
9 42
10 42
11 42
12 42
13 42
14 42
15 42
16 42
17 42
18 42
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20 42
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21 44
21 45
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22 42
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25 42
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27 42
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30 42
30 84
31 42
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32 42
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33 42
33 84
34 42
34 84
35 42
35 84
36 42
36 84
37 42
37 84
38 42
38 84
39 42
39 84
40 42
40 84
41 42
41 84
42 11
42 12
42 13
42 14
42 15
42 16
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42 18
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42 20
42 21
42 22
42 23
42 24
42 25
42 26
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42 35
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42 37
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42 41
42 42
42 84
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46 84
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48 84
49 84
50 84
51 84
52 84
53 84
54 84
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Address of Correspondent:
Graeme S. Cumming [Erratum]
Dept. Wildlife Ecology and Conservation, University of Florida , 308 Newins-Ziegler Hall , P.O. Box 110430, Gainesville, FL 32611
Phone: (352) 846-0558 Fax: (352) 392-6984 cummingg@wec.ufl.edu
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