Colonisation Process

1
Catastrophe Management and Inter-Reserve Distance
for Marine Reserve Networks
Joshua Ross
Liam Wagner
Hugh P. Possingham
Abstract We consider the optimal spacing between
marine reserves for maximising the viability of a
species occupying a reserve network. The closer
the networks are placed together, the higher the
probability of colonisation of an empty reserve
by an occupied reserve, thus increasing
population viability. However, the closer the
networks are placed together, the higher the
probability that a catastrophe will cause
extinction of the species in both reserves, thus
decreasing population viability. Using a simple
discrete-time Markov chain model for the presence
or absence of the species in each reserve we
determine the distance between the two reserves
which provides the optimal trade-off between
these processes, resulting in maximum viability
of the species.
The Department of Mathematics ARC Centre for
Complex Systems ARC Centre of Excellence for
Mathematics and Statistics of Complex Systems
The School of Integrative Biology
Introduction The design of reserve networks for
marine conservation is a contentious issue from
both theoretical and practical points of view.
One of the difficult theoretical issues in marine
reserve system design concerns the optimal
connectedness of the reserves within the network.
The closer two reserves are placed together, the
more likely it is that a population occupying one
reserve will colonise the other, increasing
population viability. However, most theories fail
to consider the possibility of catastrophic
events (Possingham et al., 2000 Shafer, 2001
Sala et al.,2002 Allison et al., 2003) the
closer two reserves are placed together the more
likely they are to be struck by the same
catastrophic event, thus decreasing viability.
Consequently, there exists a natural tension
between these two processes. Whilst this
trade-off has been identified by many authors, a
detailed theoretical investigation of its
influence and the subsequent analytic
determination of an optimal spacing, has, to the
best of our knowledge, not been undertaken (Fig
1).
This new transition matrix, with catastrophes,
colonisation and external recruitment combined,
gives has a stationary distribution for the
metapopulation which is the first eigenvector.
To construct the new overall transition matrix,
A, we first need to construct an external
recruitment matrix, R that requires only one
parameter, a, the probability an empty patch is
colonised by external recruits
Reserve 1
d
The overall model now has a Markov given by
Now that we no longer have an absorbing state our
analysis changes slightly. We use the zero patch
occupancy term in the first eigenvector to
determine the chance no patches are occupied. We
say that the metapopulation survives if the
species exists in at least one patch.
Reserve 2
Fig. 4 The probability of survival with local
extinction r 0.5, a 0.1, µ 5, a 0.1 and
b 0.025 (solid line), 0.05 (dots) and 0.1
(dashed line), as three local extinction
probabilities.
The Model
X
This model is constructed to find the optimal
spacing between two marine reserves when there is
a catastrophe. For each reserve we model the
presence or absence of the marine species. We
assume that two processes affect the presence or
absence of the species in the two-patch
metapopulation catastrophes that wipe out the
population in a single reserve and may or may not
affect both reserves, and colonisations from an
occupied reserve to an unoccupied reserve. In
our model we assume that catastrophes of mean
size µ strikes the coast at a random point and
affects both reserves with probability exp(-d/µ).
Therefore the catastrophe affects only one
reserve with probability 1-exp(-d/µ). This
formulation assumes that the probability that a
catastrophe affects both reserves decays
exponentially with increasing inter-reserve
distance. This is a reasonable assumption, in
particular for catastrophes with a radius that is
exponentially distributed in size, with an
average radius of µ. Normally a two patch
metapopulation would have four possible states.
We have reduced the state space to three by
assuming the reserves are identical and hence we
only need to keep track of whether neither
(extinction), one or both patches are occupied.
We have used basic probability theory to
construct a Markov chain for each of the
processes colonisation and catastrophes - in
this model. The state space diagrams and Markov
chain matrices are
Oil Spill
Results Ecologists and marine park managers have
for a great deal of time relied upon practical
experience and qualitative methods to design
marine reserves for conservation purposes
(Shafer, 2001). The additional inclusion of a
variety of stake holders into the process of
marine reserve design may also detract from the
formulation of a conservation management plan
that ensures the survival of a particular
species. We have provided methods that allow the
precise calculation of the optimal inter-reserve
distance. This allows decision makers to have
quick estimates for optimal inter-reserve
spacing, and thus quantitative information for
discussion with stake holders. Having found the
algebraic expression for the second eigenvalue of
a two patch presence-absence metapopulation we
can determine the optimal spacing between any two
reserves given a mean catastrophe size,
catastrophe arrival rate and local colonisation
probability. This analytic solution shows how to
optimally space marine reserves to accommodate
the possibility of catastrophe. We go on to
include the possibility of external recruitment
and local extinction events. With the expansion
of the model there is now no absorbing state and
the first eigenvector tells us the probability
that no patch has the species at equilibrium. To
calculate the optimal reserve design strategy we
find the spacing between reserves that minimises
the first value in the first eigenvector (which
is the quasi-equilibrium chance of extinction).
While this model provides insights in to optimal
spacing between reserves for a single species
it may be difficult to operate when there are
many species each with their own ability to move
and each with a different response to
catastrophes.
Mean size
Fig. 1 Schematic of an example scenario where
an oil spill has occurred along a coast line with
two defined marine reserves which are placed at
some distance apart.
The rate at which the metapopulation decays to
extinction from quasi-equilibrium is the second
eigenvalue of the matrix A the second
eigenvector defines the quasi-stationary
distribution Darroch65. We have found a method
which allows us to define an algebraic expression
for the second eigenvalue of the Markov chain.
This is graphed below as a function of the
inter-reserve spacing d. Where the second
eigenvalue is largest the extinction rate of the
metapopulation from quasistationarity is
minimised.
Fig. 3 The probability of survival with
external recruitment r 0.5, a 0.1, µ 5
and a 0.05 (solid line), 0.075 (dots), 0.10
(dashed line), as three different local
recruitment probabilities.
Local Extinction Effects
Colonisation Process
Another important aspect which needs to be
analyzed in this investigation is the chance of
local extinction with no catastrophe. We
construct another matrix which represents the
effect of local extinction (which may occur
through unfavorable local environmental
conditions) on the viability of the whole
metapopulation. We let the variable b to be the
rate of local Extinction. We will now construct
the transition matrix for local extinctions.
1
0
2
References Allison, G.W., Gaines, S.D.,
Lubchenco, J., Possingham, H.P., 2003. Ensuring
persistence of marine reserves catastrophes
require adopting an insurance factor. Ecol. Appl.
13, S8S24. Darroch, J.N., Seneta, E.J., 1965. On
quasi-stationary distributions in absorbing
discrete-time finite Markov chains. J. Appl.
Probab. 2, 88100. Day, J.R., Possingham, H.P.,
1995. A stochastic metapopulation model with
variability in patch size and position. Theor.
Pop. Biol. 48, 333360. Gilpin, M., 1992.
Demographic stochasticity a Markovian approach.
J. Theor. Biol. 154, 18. Possingham, H.P.,
Ball, I.R., Andelman, S., 2000. Mathematical
methods for identifying representative reserve
networks. In Ferson, S., Burgman, M. (Eds.),
Quantitative Methods for Conservation Biology.
Springer-Verlag, New York 291305. Sala, E.,
Aburto-Oropeza, O., Paredes, G., Parra, J.,
Barrera, J.C., Dayton, P.K., 2002. A general
model for designing networks of marine reserves.
Science 298, 19911993. Shafer, C.L., 2001.
Inter-reserve distance. Biol. Conserv. 100,
215227.
Extinction Process
Fig. 2 The probability of survival with local
extinction r 0.5, a 0.1, µ 5, a 0.1 and b
0.025 (solid line), 0.05 (dots) and 0.1 (dashed
line), as three local extinction probabilities.
1
0
2
External Recruitment
In marine reserves the few species have such a
restricted occurrence that they only exist in two
places. A logical extension of the model is to
allow for external recruitment in to the two
patch metapopulation. The inclusion of external
recruitment into our model creates a system
without an absorbing state.
1
0
2
Having constructed these matrices we can
calculate the transition matrix A for the
complete process. We assume that extinction
occurs before colonisation, and thus A E x C
(Gilpin, 1992 Day and Possingham, 1995). To
calculate the probability of survival at some
future time we may recursively evaluate Pt1
Pt x A, where Pt is a vector containing the
probabilities of being in each state at time step
t.
The final Markov chain is given by the equation
1
0
2
After constructing this Markov chain we perform
the same analysis as with the previous model and
find the spacing between reserves that maximize
the chance the population is extant.