# A Dynamic Programming Approach to Ecosystem Management
## 2 Materials and Methods
### An Issue in Populations Management  
To illustrate the application of the previously described procedure, we consider at first an example already presented and analyzed, see [30]. It describes a predator-prey system modeling the interactions between the European red fox _Vulpes vulpes_ and rodents as their prey. In addition, the parasitic tapeworm _Echinococcus multilocularis_ may infect both populations. It is possible that by contact between rodents and domestic animals, also the latter may be infected by this parasite. As a consequence, it may ultimately be harmful to humans as well. Because the parasite is difficult to eradicate, a model to study its spread has been proposed. It concentrates only on the interactions in the wilderness, not explicitly modeling the parasites [30]. The system thus explicitly describes just the dynamics of the healthy foxes \(F\), healthy rodents \(S\), infected foxes \(C\) and infected rodents \(I\) populations. Red foxes are the definitive hosts of the _Echinococcus multilocularis_. They become infected by preying on rodent intermediate hosts that ingested the parasite's eggs. The healthy rodents may get infected also by interactions with other infected rodents, or by direct contact with infected foxes, surviving their attacks, or indirectly, through fox-released environmental contamination. From [30], the model is written using mass action disease incidence, meaning homogeneous populations mixing, which gives the following system  
\[\frac{dF}{dt}=r(F+C)-mF+e(k_{2}S+k_{3}I)C+ek_{1}FS-F(b_{1}F+b_{2}C)- \lambda FI-\alpha FC+\gamma_{1}C, \tag{1}\] \[\frac{dC}{dt}=\lambda FI+\alpha FC-(m+\mu)C-C(c_{1}F+c_{2}C)- \gamma_{1}C,\] \[\frac{dS}{dt}=s(S+I)-nS-S(g_{1}S+g_{2}I)-S(k_{1}F+k_{2}C)-\theta SI -\beta SC+\gamma_{2}I,\] \[\frac{dI}{dt}=\beta SC+\theta SI-I(n+\
u)-I[(g_{3}S+g_{4}I)+( \lambda F+k_{3}C)+\gamma_{2}].\]  
The model has been completely analyzed, determining its equilibria and their feasibility and stability conditions [30]. In particular, as a result of this investigation, a graph of the system stationary points, where the thriving populations are indicated without explicitly writing their equilibrium levels, linked by transcritical bifurcations, has been obtained. The parameters involved in the latter are the four populations' natural and disease-induced mortalities, \(m\), \(n\), \(\mu\), \(\
u\) for healthy foxes and rodents, and infected foxes and rodents, respectively. Figure 1 shows the graph of the equilibria and the transcritical bifurcations relating them. In what follows, we assume that the current state of the system is the coexistence endemic equilibrium \((F,C,S,I)\), which is assumed to be stable. This corresponds to the actual conditions in the wild, where the two animal populations thrive, with the parasite being endemic among them. Our goal is to efficiently reach a pest-free equilibrium, either the parasite and rodent-free point \((F,0,0,0)\), removing infected foxes and both healthy and infected rodents at the minimum cost. Alternatively, we will also seek the parasite-free equilibrium \((F,0,5,0)\), where the disease is eradicated, but healthy rodents thrive. This problem can be rewritten as a minimum cost flow problem on the network built using the system equilibria as nodes: we consider the current state of the population system as the supply node and the desired state as the demand node. The arcs in the graph connect the equilibria that are linked by transcritical bifurcations. By acting on the bifurcation parameters, namely the mortalities, it would be possible to shift stability of one equilibrium and move to an adjacent node. Concretely, the operation of changing the parameter values requires some human operation. Therefore, we can associate a positive quantity to each arc, representing the cost of the corresponding activity. Our objective is to minimize the overall cost of the necessary operations needed to bring the system from its current status to the final desired configuration.