\subsection{Wolf-Deer Predator-Prey Model}
The predator-prey relationship between
the wolves and deer can be modelled by both modifying (\ref{eq:w1}) to
include a ``preytaxis'' term, describing how the wolves move
towards herds of deer, and a mortality term, describing how the pack
may starve in the absence of deer, and by including a second equation
describing the deer dynamics. We model the deer population
as exhibiting density-dependent logistic growth
and being subject to predation.
Thus we have:
\begin{eqnarray}
\lefteqn{\mbox{Rate of change in expected density of wolves }}\nonumber\\
&=&
\mbox{Rate of change due to flux of wolves towards the den (summer only)}\nonumber\\
&+&
\mbox{Rate of change due to dispersal of wolves away from high density} \nonumber\\
& &
\mbox{regions in search of food and other resources} \nonumber\\
&+&
\mbox{Rate of change due to flux of wolves towards the white-tailed deer}\nonumber\\
&-&
\mbox{Rate of change due to wolf starvation resulting from limited deer}\nonumber
\end{eqnarray}
\begin{eqnarray}
\tpa{u}&=&\nabla\cdot\left[c_u
\frac{(\ul{x}-\ul{x}_u)}{|\ul{x}-\ul{x}_u|}
u\right]
+\nabla\cdot\left[D_un_1(u/u_0)\nabla u\right] \nonumber \\
&+&\nabla\cdot\left[us_un_3(h/h_0)\nabla h\right]
-r_u u \left[ {uh_0}/{hu_0}\right] ;\label{eq:w1b}
\end{eqnarray}
\begin{eqnarray}
\lefteqn{\mbox{Rate of change in density of white-tailed deer}}\nonumber\\
&=&
\mbox{Flux away from the high density regions of deer (decreases in}\nonumber\\
& &
\mbox{ winter)}\nonumber\\
&-&
\mbox{Rate of mortality from wolves}\nonumber
\end{eqnarray}
\begin{equation}
\tpa{h}=\nabla\cdot\left[D_h n_4(h/h_0)\nabla{h}\right]
-e_u u \frac{h/h_0}{1+bh/h_0}. \label{eq:hb}
\end{equation}
Here $s_u$ and $D_h$ scale
the ``preytaxis'' rate and the deer diffusion rate, respectively.
The rate of starvation per wolf is inversely proportional to the
amount of available prey per wolf $(hu_0/uh_0)$ and is scaled by $r_u$.
The decline of deer due to wolf predation is given by a Holling `type 2'
functional response (see, for example, Begon and Mortimer (1981))
and is scaled by $e_u$. The per predator saturation
rate is given by $e_u/b$.
The nonlinear response
in wolf movement to the changing deer density is described
by $n_3(h/h_0)$, the density-dependent diffusion of deer is
described by $n_4(h/h_0)$
and the qualitative forms for $n_3(\cdot)$ and $n_4(\cdot)$ are
as in Figure 3. The `preytaxis' response ($n_3(\cdot)$) is most sensitive at
very low prey levels, when the chance of finding deer is lowest,
and $n_4(\cdot)$ indicates that the rate of deer dispersal increases as the deer
density increases.
Based on our discussion in Section 3, we expect that $D_h$
will be reduced in the winter months, when the deer group together in
a yard, and will be increased in the summer months when the
deer disperse.
Equations (\ref{eq:w1b}) and (\ref{eq:hb}) describe the spatial dynamics
of wolf-deer interactions in the absence of inter-pack buffer zones.
The spatial dynamics of this system will be analysed and compared with the dynamics
of composite system, where both inter-pack competition and predation are
included (Section 4.2.4, below) (see also Section 5.1). We anticipate that the
comparison will help to explain the role that wolf territoriality plays in
regulating the wolf-deer interactions.
\subsection{Composite model}
When we include both inter-pack competition and predation,
the complex system of nonlinear equations [(\ref{eq:w1a})--(\ref{eq:hb})],
given in a nondimensional form, is
\begin{eqnarray}
\tpa{u}&=&\nabla\cdot\left[
\gamma_u\frac{(\ul{x}-\ul{x}_u)}{|\ul{x}-\ul{x}_u|}
u\right]
+\nabla\cdot\left[n_1(u)\nabla u\right]\nonumber\\
&+&\nabla\cdot\left[ u\tau_un_2(q)\nabla q\right]
-\nabla\cdot\left[ u\sigma_un_3(h)\nabla h\right]\nonumber\\
&-&\kappa_uuv-\rho_uu^2/h\label{eq:m1nd1}\\[1ex]
\tpa{v}&=&\nabla\cdot\left[\gamma_v
\frac{(\ul{x}-\ul{x}_v)}{|\ul{x}-\ul{x}_v|}
v\right]
+\nabla\cdot\left[\delta_v n_1(v)\nabla v\right]\nonumber\\
&+&\nabla\cdot\left[ v\tau_vn_2(p)\nabla p\right]
-\nabla\cdot\left[ v\sigma_vn_3(h)\nabla h\right]\nonumber\\
&-&\kappa_vuv-\rho_vv^2/h\label{eq:m1nd2}\\[1ex]
\tpa{p}&=&u\left(\lambda_p+\mu_pq\right)-p\label{eq:m1nd3}\\[1ex]
\tpa{q}&=&v\left(\lambda_q+\mu_qp\right)-\phi_pq\label{eq:m1nd4}\\[1ex]
\tpa{h}&=&\nabla\cdot\left[\delta_h n_4(h)\nabla{h}\right]
-\left[\epsilon_u u + \epsilon_v v\right] \frac{h}{1+bh}.\label{eq:m1nd5}
\end{eqnarray}
%where $n_1(\cdot),\ldots,n_4(\cdot)$ are describe the nonlinear dependence of
%the flux on the local concentrations. An appropriate form may be the power law:
%\begin{equation}
%n_i(a)=a^{\alpha_i},\;\;\;\alpha_i>0.
%\end{equation}
Analysis of (\ref{eq:m1nd1})--(\ref{eq:m1nd5}), with seasonal differences in
$\gamma_u$, $\gamma_v$, $\delta_h$, $\lambda_p$, $\lambda_q$, $\mu_q$
and $\mu_p$, will help to explain yearly cycles in wolf territories and deer
patterns (see Section 5.1).
Models in this section address the {\em behavioral} interactions which determine
spatial {\em distributions} of wolves and deer over short time scales (typically,
scales of several months). In the next section we consider the {\em ecological}
interactions which determine the {\em abundance}
of wolves and deer over longer time scales (typically, scales of several years).
\section{Model 2: Ecological Model}
In this section we derive discrete mathematical models
describing the wolf-deer ecological interactions outlined for the
second model in Section 4.1.
To keep our presentation relatively simple, we consider two different models;
the first includes a spatial refuge, but no migration, while the second includes
migration, but no spatial refuge. It is possible to consider a model which includes
both by incorporating features from both the models described here.
We again give the model equations in words prior to their mathematical formulation.
First, we consider the model that has a spatial refuge. We assume that
there can be a flux of deer from the refuge into non-refuge areas at the
beginning of each time period.
State variables are:
\begin{eqnarray}
U&=&\mbox{Number of wolves in the pack},\nonumber\\
H&=&\mbox{Number of deer in the wolf territory},\nonumber\\
R&=&\mbox{Number of deer in the refuge}.\nonumber
\end{eqnarray}
This scenario may be described
by the following system of difference equations:
\begin{eqnarray}
\lefteqn{\mbox{Number of wolves at the beginning of the year}}\nonumber\\
&=&
\mbox{Number of adult wolves from previous year}\nonumber\\
&\times&
\mbox{Probability of an adult wolf surviving the year}\nonumber\\
& &\mbox{(based on deer availability)}\nonumber\\
&+&
\mbox{Number of pups born in the spring}\nonumber\\
&\times&
\mbox{Probability of a juvenile wolf surviving the year}\nonumber\\
& &\mbox{(based on deer availability and wolf pack size)}\nonumber
\end{eqnarray}
\begin{equation}
U_{t+1}= U_ts_1^A(H_t)+b_1(H_t) s_1^J(H_t,U_t);\label{eq:SR1}
\end{equation}
\begin{eqnarray}
\lefteqn{\mbox{Number of deer in wolf territory at the beginning of the year}}\nonumber\\
&=&
\mbox{[Number of adult deer in wolf territory from previous year}\nonumber\\
&+&
\mbox{Number of deer moving from the refuge into the wolf territory]}\nonumber\\
&\times&
\mbox{Probability of adult deer surviving the year}\nonumber\\
& &\mbox{(based on the wolf population)}\nonumber\\
&+&
\mbox{Number of fawns born in wolf territory in the spring}\nonumber\\
&\times&
\mbox{Probability of juvenile deer surviving the year}\nonumber\\
& &\mbox{(based on the wolf population)}\nonumber
\end{eqnarray}
\begin{equation}
H_{t+1}=\left[ H_t + m(H_t,R_t)\right] s_2^A(U_t)+b_2(H_t)s_2^J(U_t);\label{eq:SR2}
\end{equation}
\begin{eqnarray}
\lefteqn{\mbox{Number of deer in refuge areas at the beginning of the year}}\nonumber\\
&=&
\mbox{Number of deer in the refuge from previous year}\nonumber\\
&+&
\mbox{Number of fawns born in the refuge in the spring}\nonumber\\
&-&
\mbox{Number of deer moving from the refuge into wolf territory}\nonumber
\end{eqnarray}
\begin{equation}
R_{t+1}=R_t+b_2(R_t)-m(H_t,R_t).\label{eq:SR3}
\end{equation}
Here $b_1(H_t)$ and $b_2(H_t)$ are the birth rates for wolves and deer, respectively,
$s_1^J(H_t,U_t)$ and $s_2^J(U_t)$ are the survival rates for juvenile
wolves and deer, respectively,
$s_1^A(H_t)$ and $s_2^A(U_t)$ are the survival rates for adult
wolves and deer, respectively,
and $m(H_t,R_t)$ describes the flux between refuge and non-refuge areas.
Note that the birth rate for wolves ($b_1$) differs from many classical formulations
in that it depends on the available deer population ($H_t$) rather than the pack
size ($U_t$). This arises from the fact that typically only the alpha pair mates
({\em c.f.} Sections 2 and 3) and that the number of pups depends upon the health
of the alpha pair, which in turn depends upon the available food (i.e.\ the number of
deer). Hover, the survival of juvenile wolves depends upon the number of adult
woves that can assist with feeding ($s_1^J$ depends upon $U_t$ as well as $H_t$).
We assume that the mortality of deer in refuge areas
is negligible. If density-dependent deer mortality were significant, then the terms
$R_t$ and $b_2(R_t)$, on the right hand side of equation (\ref{eq:SR3}),
could be replaced by $R_t/(1+aR_t)$ and $b_2(R_t)/(1+aR_t)$,
and similarly for equation (\ref{eq:SR2}) (see, for example Hassell (1975)).
Second, we consider the model that permits migration, but has no refuges.
Here we have two further variables:
\begin{eqnarray}
V&=&\mbox{Size of wolf pack number 2},\nonumber\\
G&=&\mbox{Size of deer herd number 2}.\nonumber
\end{eqnarray}
We assume that deer herd number 1 (whose size is represented by $H$) spends
the summer in the territory of wolf pack number 1 (whose size is represented
by $U$) and winter in the territory of wolf pack number 2 (whose size is represented
by $V$). The converse holds true for deer herd number 2
(whose size is represented by $G$).
This scenario may be described by the following system of difference equations:
\begin{eqnarray}
\lefteqn{\mbox{Number of wolves after the fall migration of deer}}\nonumber\\
&=&
\mbox{Number of adult wolves at the end of last winter}\nonumber\\
&\times&
\mbox{Probability of an adult wolf surviving the summer}\nonumber\\
& &\mbox{(based on deer availability)}\nonumber\\
&+&
\mbox{Number of pups born in the spring}\nonumber\\
&\times&
\mbox{Probability of a juvenile wolf surviving the summer}\nonumber\\
& &\mbox{(based on deer availability and wolf pack size)}\nonumber
\end{eqnarray}
\begin{eqnarray}
U_{t+1/2}&=& U_ts_1^{AS}(H_t)+b_1(H_t) s_1^{JS}(H_t,U_t),\label{eq:SM1}\\
V_{t+1/2}&=& V_ts_1^{AS}(G_t)+b_1(G_t) s_1^{JS}(G_t,V_t);\label{eq:SM2}
\end{eqnarray}
\begin{eqnarray}
\lefteqn{\mbox{Number of deer after the fall migration}}\nonumber\\
&=&
\mbox{Number of adult deer that emigrated in the spring}\nonumber\\
&\times&
\mbox{Probability of an adult deer surviving the summer}\nonumber\\
& &\mbox{(based on the number of wolves)}\nonumber\\
&+&
\mbox{Number of fawns born in the spring}\nonumber\\
&\times&
\mbox{Probability of a juvenile deer surviving the summer}\nonumber\\
& &\mbox{(based on the number of wolves)}\nonumber
\end{eqnarray}
\begin{eqnarray}
H_{t+1/2}&=&H_t s_2^{AS}(U_t)+b_2(H_t)s_2^{JS}(U_t),\label{eq:SM3}\\
G_{t+1/2}&=&G_t s_2^{AS}(V_t)+b_2(G_t)s_2^{JS}(V_t);\label{eq:SM4}
\end{eqnarray}
\begin{eqnarray}
\lefteqn{\mbox{Number of wolves after the spring migration of deer}}\nonumber\\
&=&
\mbox{Number of wolves at the end of last summer}\nonumber\\
&\times&
\mbox{Probability of a wolf surviving the winter}\nonumber\\
& &\mbox{(based on deer availability)}\nonumber
\end{eqnarray}
\begin{eqnarray}
U_{t+1}&=&U_{t+1/2}s_1^W(G_{t+1/2}),\label{eq:SM5}\\
V_{t+1}&=&V_{t+1/2}s_1^W(H_{t+1/2});\label{eq:SM6}
\end{eqnarray}
\begin{eqnarray}
\lefteqn{\mbox{Number of deer after the spring migration}}\nonumber\\
&=&
\mbox{Number of deer that emigrated in the fall}\nonumber\\
&\times&
\mbox{Probability of a deer surviving the winter}\nonumber\\
& &\mbox{(based on the number of wolves)}\nonumber
\end{eqnarray}
\begin{eqnarray}
H_{t+1}&=&H_{t+1/2}s_2^W(V_{t+1/2}),\label{eq:SM7}\\
G_{t+1}&=&G_{t+1/2}s_2^W(U_{t+1/2}).\label{eq:SM8}
\end{eqnarray}
Here $b_1(\cdot)$ and $b_2(\cdot)$ are as before,
$s_1^{AS}(\cdot)$, $s_1^{JS}(\cdot)$ and $s_1^W(\cdot)$ are the survival rates of
adult and juvenile wolves in summer and wolves and winter, respectively, and
$s_2^{AS}(\cdot)$, $s_2^{JS}(\cdot)$ and $s_2^W(\cdot)$ are the survival rates of
adult and juvenile deer in summer and deer in winter, respectively.
The time step $t+1/2$ is just after the fall deer migration and the
time steps $t$ and $t+1$ are just after the spring migration of deer.
Substitution of [(\ref{eq:SM1})--(\ref{eq:SM4})] into
[(\ref{eq:SM5})--(\ref{eq:SM8})] yields a system of four equations.
Very little analytical work has been done on large discrete
systems such as these and we anticipate that the mathematics
will be interesting and challenging.
These two systems of difference equations [(\ref{eq:SR1})--(\ref{eq:SR3})
and (\ref{eq:SM1})--(\ref{eq:SM8})] describe the ecological interactions
of wolves and deer in the presence of a deer refuge and with deer migration,
respectively. Mathematical and numerical analysis will used to determine
the effects that refuges and deer migration have upon the
nature of wolf-deer interactions and upon the stability of wolf and deer
population levels (see Section 5.2).