# ELE1004SA ```Appendix A
The purpose of this appendix is to solve the classical diffusion problem under the conditions
expected for the BDFFP data, except that we assume uniform sampling. Since our MLE analysis
uses all data aggregated irrespective of recapture delay, what we actually observe is not the
amplitude dispersal kernel itself but the time–integrated amplitude dispersal kernel, which will
differ from the dispersal kernel itself. Therefore, we pay special attention to T, the timescale over
which observations are collected, e.g. whether we wait significantly longer than the lifetime of
organisms or not. We demonstrate that, at large radii, the dependence ranges from a Gaussian tail
(at short T) to an exponential tail (at large T). Power–law tails are thus not expected to arise
under classical diffusion.
The Distribution of Distances Traveled Using Classical Assumptions
Suppose an organism is released at t=0 at the origin. What is the probability that the animal is
caught in a trap at a distance r away any time between t1 and t2? Let us assume the probability of
getting caught at this distance depends only on the number of traps this far away and that this
probability is p(C|r)=w(r). Since r is a continuous variable, we have a probability density:
f (C, ri )  p(C | r ) f (r )  2 r  w(r )  f (r )  w(r ) g (r )
…[A1]
The functions f(r) and g(r) are related to the dispersal kernel and the amplitude kernel
respectively. Here, were are assuming that w(r) is the density of traps at a distance r while f(r) is
the amount of time that the organism spends at this distance away between t1 and t2 before dying,
this can be modeled by the classical diffusion approach. This probability is given by:
t2
f (r )   f (r ; t )exp[ t /   ]dt
…[A2]
t1
The exponential factor in (A2) arises because the organism must have survived death; assume
the probability of this is exp[–t/] where  represents the average lifetime of the organism. The
term f(r;t) is the dispersal kernel . The associated amplitude kernel is simply the dispersal kernel
integrated around the annulus of radius r, namely g (r ; t )  2 r  f (r ; t ) . The dispersal kernel is
here the classical Gaussian diffusion form (Turchin 1998):
f (r ; t ) 
 r2 
1
exp  

4 Dt
 4 Dt 
…[A3]
Let us calculate the marginal density of captures given by [A2] for a system with uniform trap
density. If we put together Equations [A1]–[A3] then we have the integral:
t
  r2
r 2
t   dt
f (C , r )  w(r ) 
exp




 

2 D t1
  4 Dt     t
…[A4]
If we were only interested in the case where t1t2, then the integral wouldn’t change much, and
the radial dependence would have the form of a Rayleigh distribution, which at large radii decays
like the Normal distribution.
More often (this is the case in our analysis) we are considering the sum of all captures starting at
t1=0 up to a total observation time t2 =T, so we must integrate:
f (C , r ) 
T
 
w0 r
t   dt
exp    D   



2D 0
  t     t
…[A4]
Here, w(r)=w0 because of the assumed uniform density of traps. Also, we have defined the
diffusion time  D  r 2 / 4 D . This leads to two separate cases, depending on how long we wait.
Case I: Short observation times: In the special case where T is small, the death of organisms
plays a negligible role in the dynamics and the second term in the exponential in (A4) is
negligible compared to the first. Then the integral becomes:
wr
   dt
f (C, r )  0  exp  D 
2D 0
 t  t
T
…[A5]
1
The result of this integration is expressed in terms of the exponential integral function E(x),
which for large values of x can be approximated by a simpler exponential function E (x)exp(–
x)/x so that we have:
f (C , r ) 
 r2 
w0r  r 2  2w0T
E

exp



2 D  4 DT 
r
 4 DT 
…[A6]
This is still dominated by Gaussian dependence at large values of r.
 D  , we must take into account both terms in
Case II: Long observation times: When T
the argument of the exponential, we cannot ignore the first term because this will always be large
for some of the range of integration. So we evaluate the integral A4, using the change of variable


  ln t /  D  , for which it becomes:

f (C , r ) 
w0 r
 exp  2 p cosh   d
2 D 

Here, the upper limit of the integral is   ln T /  D 
 and the parameter p 
…[A7]
 D /   . Note
that if T is sufficiently large, the upper limit is essentially infinite. The peak value of the
integrand is at =0 and rapidly decays when ||&gt;1, so we can use a Taylor expansion of the cosh–
function about zero to arrive at an acceptable approximation. The expansion is cosh(x)=1+x2/2!+
x4/4!+… and if we keep just the first two terms, the integral in (A7) becomes:

  2 

 exp 2 p 1  2  d  p exp  2 p 

When we substitute this into the (A4) and convert back to the parameters D and , whence:
2
w 
f (C , r )   0 
D
3/ 4
r
2
exp  r / D  
…[A8]
This says that in classical diffusion, the density of captures should (given uniform trap density at
any radius) be dominated by exponential decay with radius. The scale of this exponential is given
by D  , the geometric mean of the lifetime and the diffusion constant.
3
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