Department of Mathematics

IDEA: Internet Differential Equations Activities

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 tex2html_wrap_inline136 and tex2html_wrap_inline138 represent the associated annual net rates of change [head/square mile/year]. The following pair of differential equations models tex2html_wrap_inline136 and tex2html_wrap_inline138 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:

eqnarray27

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

eqnarray37

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., tex2html_wrap_inline172 as tex2html_wrap_inline174 ), 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:

equation57

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 tex2html_wrap_inline226 (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

eqnarray85

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 tex2html_wrap_inline242 in (7) and solving for x in terms of y, is given by

equation93





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

equation100

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.


Figure 1. Zero-trapping Phase Plane


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 tex2html_wrap_inline304 and tex2html_wrap_inline306 and tex2html_wrap_inline308 , and tex2html_wrap_inline310 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 tex2html_wrap_inline324 [see (6)], and the per capita growth rate for y as tex2html_wrap_inline328 [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 tex2html_wrap_inline342 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 tex2html_wrap_inline372 .

Assume that tex2html_wrap_inline372 represents a 100% annual trapping rate (i.e., tex2html_wrap_inline376 ), 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 tex2html_wrap_inline382 yields the implicit expression for the nullcline

equation122

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 tex2html_wrap_inline398 and xi(0) = 0 when tex2html_wrap_inline402 . The nullcline xi is pictured along with yi below. What is the biological interpretation of tex2html_wrap_inline408 ? 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 tex2html_wrap_inline422 hd/sq mi/year. The applet below may be used to generate a phase diagram for the trapping scenario.

With the advent of HTML5, Javascript is now ready for prime time for mathematical applications. There are new Javascript demos illustrating how we might use interactive web objects to help students learn Calculus.

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This project is supported, in part, by the National Science Foundation. Opinions expressed are those of the authors, and not necessarily those of the Foundation.
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