SMALL MAMMAL DISPERSION
Ray Huffaker, Thomas LoFaro, and Kevin Cooper
Washington State University
Setting the Scene: Beavers, once hunted in open access for their pelts, were saved from extinction in the middle of this century by regulations controlling trapping season, method and numbers. Under this protection, the beaver population has rebounded in many regions of the country and has caused significant damage to valuable timber and agricultural land. Trapping is most effective in controlling beavers, whose primary nuisance is tree-cutting on privately-held timber land.
A trapping strategy that disregards the possible migratory
behavior of beavers in neighboring ``uncontrolled'' (i.e., zero
trapping) land parcels in filling the vacuum created by trapping
in the ``controlled'' parcel, can be as futile in practice as
attempting to dig a hole in fine-grain sand. We formulate a
two-equation system of differential equations to model this
phenomenon according to the recently formulated ``social-fence''
hypothesis of small mammal dispersion. This hypothesis can be
viewed as the ecological analog of osmosis: Beavers from an
environmentally superior habitat are posited to diffuse through a
social fence to an inferior but less-densely populated habitat
until the pressure to depart (``within-group aggression'') is
equalized with the pressure exerted against invasion
(``between-group aggression''). This is termed ``forward
migration.'' Assuming that the controlled parcel is a superior
habitat, the owner must be concerned with the ``backward
migration'' that occurs when the superior parcel becomes less
densely populated through trapping.
Rate Equations: Let X
and Y represent nonnegative population densities
[head/square mile] of beavers in the controlled and uncontrolled
parcels respectively; and let 
and 
represent the associated annual net rates of change
[head/square mile/year]. The following pair of differential
equations models and as the difference between the rates of net growth
(i.e., birth rate minus the death rate), dispersion, and, in the
case of X, trapping:
where P [1/year] represents the per capita annual trapping rate of X. Thus, PX represents the total animals trapped each year.
F0(X) and F2(X) are logistic per capita (proportional) population growth rates for X and Y respectively with units [1/year] and are given by
where RX [1/year], KX
[head/square mile], RY [1/year], and KY
[head/square mile] are nonnegative constants. As the population
density X approaches zero in the controlled parcel, the
net proportional growth rate approaches RX
(i.e., 
as 
), which is
called the intrinsic growth rate [1/year]. Alternatively, as X
approaches KX, the net proportional
growth rate decreases toward zero due to the negative impacts of
crowding. Thus KX is the environmental
carrying capacity of the controlled parcel for beavers. The parameters
RY and KY
are interpreted analogously for the uncontrolled parcel.
The total dispersion flux term, F1(X,Y), is a mathematical representation of the social-fence hypothesis (discussed above), which attempts to explain the dispersion of a small-mammal population between two adjacent parcels:
where M and -M [head/square mile/year] are the marginal disperse rates with respect to X/KX and Y/KY, respectively. Whenever X constitutes a larger fraction of its carrying capacity than Y (i.e., X/KX > Y/KY ), within-group aggression of X is assumed to be greater than between-group aggression, and F1(X,Y) acts as a disperse valve allowing individuals to ``forward'' migrate from X to Y, i.e., F1(X,Y) > 0. However, as trapping decreases the population pressure on carrying capacity in X, the disperse valve can become unidirectionally open for individuals to ``backward'' migrate from Y to X, i.e., F1(X,Y) < 0.
The number of system parameters (i.e., RX,
RY, KX, KY,
M, P) involved in solving system (1)-(2) can be decreased
from six to four by making the above quantities dimensionless. We
do this by letting x = X/ KX (the
fraction of carrying capacity in the controlled parcel), y =
Y/ KY (the fraction of carrying capacity in
the uncontrolled parcel), and 
(scaled time variable). Scaled parameters are m =
M/ RXKX (the
scaled dispersion parameter), p = P/ RX
(the scaled trapping parameter), r = RY
/ RX (the comparison of intrinsic growth
rates in both patches), and k = KX /
KY (the comparison of carrying capacities
in both patches).
Problem 1: Use these new variables to show that the dimensionless model is
Equations (6) and (7) form the two equation system whose
qualitative solution is outlined below using phase-diagram
techniques. We will study the solution of system (6)-(7) as the
dimensionless trapping rate p is increased from zero.
Naturally-Regulated Dynamics:
Consider first the ``naturally-regulated'' dynamics occurring
when no trapping occurs in the timber-damaged parcel. This
situation is modeled by setting p = 0 in (6).
Problem 2: Show that the nullcline
yi, derived by setting 
in (7) and solving for x in terms of y,
is given by
Problem 3: Show that the nullcline
yi is a parabola whose vertex occurs at a positive level
of y when r-km is positive, and at a
negative level when r-km is negative. Conclude that
yi'(0) < 0 in the first case and that yi'(0)
> 0 in the second. What is the biological interpretation of r-km?
Problem 4: Show that setting x'=0 in (6) with p=0 yields the implicit expression for the nullcline
associated with the controlled parcel. Denote this nullcline
by xi. Since only positive values of x are
biologically relevant, equation (9) can be solved for x
in terms of y giving us a function x = xi(y).
Do this to show that xi is a monontonically-increasing,
concave-down function. Show that xi(0) > 0 when the
scaled migration rate (m) is less than one and xi(0)
= 0 when m > 1. The nullcline xi is pictured
along with yi in the figure below. Note that the vertical
axis is labeled x and the horizontal axis y.
Problem 5:
The naturally-regulated phase plane can assume a wide range of
possible steady-state configurations (i.e., intersections of xi
and yi) depending on the sign of r-km and
how m compares to 1. Use the baseline parameter
values 
and 
and 
, and 
to plot the nullclines x=yi(y)
and x = xi(y). (Huffaker, Bhat and Lenhart,
``Optimal Trapping Strategies for Diffusing Nuisance-Beaver
Populations,'' Natural Resource Modeling 6 (1992): 71-98.
The baseline nullclines generate a dual steady-state
configuration. One steady state occurs at the carrying capacities
of the two parcels, (x,y) = (1,1), and the other
occurs at the origin. (Nullclines graphed with other parameters
can be shown to maintain these two steady states.) The nullclines
divide the phase plane into four isosectors
(see the figure above).
Problem 6: Calculate the
relevant eigenvalues to show that the equilibrium at the origin
is a saddle-point. Calculate the relevant eigenvalues to show
that the equilibrium at the carrying capacities is a stable-node
equilibrium. Use the applet below to generate a phase diagram
numerically using the baseline parameters.
Problem 7: Copy the diagram from Problem 4 by hand, and supply the directions of motion in each isosector. For example, the arrow pointing northward in isosector II indicates increasing values of x over time. The arrow pointing eastward in isosector II indicates increasing values of y over time.
The figure from Problem 4 is helpful in understanding the dynamics
associated with various regions of the phase plane. The figure
denotes the per capita growth rate for x as 
[see (6)], and the per capita growth rate
for y as 
[see (7)]. Three dashed lines are superimposed on
the nullclines in the figure to divide phase space further
into six areas. The areas bounded by the lines x = 1 and y
=1 (II and III) are characterized by positive per capita growth
rates for both parcels since each population is below carrying
capacity. Areas above x = 1 (I, IV, and V) produce
negative growth rates for x since the population is above
carrying capacity. Areas to the right of y = 1 (IV, V and
VI) produce negative growth rates for y since the
population is above carrying capacity. The dashed line running
from the origin through the nullclines at carrying capacity is
the zero-dispersion line (zdl), that sets the dispersion flux
term 
to zero, i.e., x = y.
Population levels above the zdl (I, II, and VI) open up the
social fence for forward migration from x to y,
while levels below (III, IV and V) create backward migration from
y to x.
Consider, for example, the dynamics in area II which is
bounded above by x=1 and below by zdl. Growth rates are
positive in both parcels since each population is below carrying
capacity. The social fence is open for forward migration from x
to y since population levels are above the zero-dispersion
line. Hence, x enjoys a positive growth rate, but suffers
emigration losses. Initial levels of x above xi
initially decrease over time because emigration is greater than
growth each period. However, once the population falls below xi,
the growth rate overwhelms the emigration rate, and the
population begins to increase. Conversely, positive growth rates
work together with immigration gains to increase y.
Positive Trapping Rates:
Consider now the impact of a nonzero trapping rate in the
controlled parcel x. The system of differential equations
governing the evolution of beaver populations when trapping
occurs in the controlled parcel is given by system (6)-(7) where
p is set at fixed rate 
.
Assume that represents a 100% annual
trapping rate (i.e., 
), and that all other parameters are held at
baseline values. The nullcline for the uncontrolled parcel (yi)
remains the same as in the zero-trapping case.
Problem 8: Show that
setting x'=0 in (6) with 
yields the implicit expression for the nullcline
associated with the controlled parcel. Denote this nullcline
by xi. Since only positive values of x are
biologically relevant, equation (10) can be solved for x
in terms of y giving us a function x = xi(y).
Do this to show that xi is a monontonically-increasing,
concave-down function.
Problem 9: Show that xi(0)
> 0 when 
and xi(0) = 0 when 
. The nullcline xi is pictured
along with yi below. What is the biological interpretation
of 
? Draw the new isoclines
and describe the direction of motion in each isosector. Show that
there continues to be a pair of equilibria, one at the origin
that is a saddle, and a second in the first quadrant that is a
stable-node.
Increasing the fixed trapping rate from zero shifts xi
downward. This drives the positive population equilibrium toward
the origin and below the zero dispersion line.
Therefore, the controlled parcel attracts backward migrants from
the uncontrolled parcel at a sustained level each year.
Problem 10: Using the baseline parameters given above, the positive population equilibrium is (y, x) = (.21, .06) and is a stable node.
Thus, a 100% sustained trapping rate drives down the
controlled population x to about 6% of its carrying
capacity, and the uncontrolled population y to about 21%
of its carrying capacity. The sustained backward migration rate
from y to x [see equation (1)] is 
hd/sq mi/year. The applet below may be
used to generate a phase diagram for the trapping scenario.
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