4. Farmers-Bandits interaction under fixed authority

As discussed in detail in the following Section, assuming the third state variable z to be an parameter will be a great help in understanding the behavior of system (2) when R->0, especially when system (2) is assumed to be composed by a fast subsystem (x,y) and a slow subsystem (z).

Now let us consider the subsystem (x,y). The (nonnegative ) equilibria of this subsystem are given by:

The trivial equilibria (3a) and (3b) collide at z=1/H (for z>1/Hthe state variable x in (3b) is negative and hence meaningless), while the collision of a nontrivial solution of (3c) with (3b) can be found by letting , obtaining

By evaluating the Jacobian matrix

of system (2a-b) in correspondence of (3a) and (3b), it can be checked that (3a) is unstable for z<1/H and stable for z>1/H, while (3b) is stable for and unstable for . On the other hand, the complete analytical discussion of the number of (nonnegative) solutions of (3c) and of their stability is extremly complex, and therefore beyond the scope of this work. That is why a numerical analysis has been performed for a fixed set of parameters in order to gain deeper insight into the system behavior [FFP93].
The following parameters have been fixed to the given values:

E is left free and will be used as bifurcation parameter. These values were chosen by Feichtinger [FFP93] when they introduced the investigated model of the system, and turned out to be particularly suited to evidence the relevant phenomena characterizing the behavior of the system.

In the following figure the bifurcation diagram of the nontrivial equilibria of the subsystem (x,y) is shown in the parameter space (z,E).

The curves (with the exception of HO, see below) shown in the above figure have been derived by Feichtinger et al. [FFP93] with an interactive PC program (LocBif, see [Khi93]).

In detail, the above bifurcation diagram contains the following:

• The curve HF, marking a Hopf bifurcation: for (z,E) on this curve, the Jacobian J has eigenvalues . By crossing HF from below to above a nontrivial equilibrium of the subsystem (2a-b) becomes unstable and a stable limit cycle is born.



• The curve SN, marking a saddle-node bifurcation: for (z,E) on this curve, J has an eigenvalue . The second eigenvalue is on the branch SN-, and on SN+. By crossing SN from above to below two nontrivial equilibria of the subsystem (2a-b) collide and therefore disappear. On the branch SN- these two are a stable node and a saddle, on the branch SN+ they are an unstable node and a saddle.



• The point DZ, marking a double-zero codimension-2 bifurcation: for (z,E) on this point, J has eigenvalues . The analysis of the normal form of this bifurcation, e. g. that done by Wiggins [Wig90] predicts the existence of a curve HO, rooted at point DZ, marking a homoclinic bifurcation: for (z,E) on this curve, the subsystem (2a-b) has a homoclinic orbit. Since HO cannot be derived through local bifurcation analysis, as pointed out by Feichtinger et. al. [FFP93], its location in the (z,E) plane has been approximately derived on the basis of repeated simulations.



• The curve TC, marking a transcritical bifurcation, and which is corresponding to : for (z,E) on this curve, a nontrival equilibria of (2a-b) collides with (3b).

The analysis of the above given bifurcation diagram in the (z,E) space, allows us to derive the bifurcation diagram of the subsystem (2a-b) in the space (x,y,z) (where we must not forget, that z is a parameter) for fixed values of E.
The next four figures show such diagrams, corresponding to the values E1, E2, E3, E4 evidenced in the above bifurcation diagram in the (z,E) space. All these diagrams are qualitative diagrams, both because the curves have been deformed for the sake of illustration, and because the dimension of the limit cycle cannot obviously be ascertained from the above bifurcation diagram.
Notice that all four bifurcation diagrams above have a limit cycle at z=0, since for all i (the case is less interesting since the two-dimensional subsystem (x,y) has not a limit cycle at z=0, and can be easily discussed by the reader).

bifurcation diagram for the value E1 in the space (x,y,z)

In the above diagram (E = E1) the stable limit cycle existing at z=0 shrinks by increasing the value of z, and when reaching the value it becomes a stable equilibrium, which then collides with an unstable one when increasing z even further to the value .
If now then z is decreased, this unstable equilibrium collides at (on the plane y=0) with the trivial one (3b).

bifurcation diagram for the value E2 in the space (x,y,z)

This bifurcation diagram given for an value of E equal to E2, is very similar to the above one, the first in this series of bifurcation diagrams in the space (x,y,z). It differs from the first one only because . This will be very interesting, as we will see later on.

bifurcation diagram for the value E3 in the space (x,y,z)

In the above diagramm (E = E3), the system (2a-b) has homoclinic orbits for two different values and .

bifurcation diagram for the value E4 in the space (x,y,z)

Finally, in this diagram given above (E = E4), the system (2a-b) has only one homoclinic orbit, and at there is a saddle-node collision of two unstable equilibria.

This four bifurcation diagrams given above are of great help in understanding the behaviour of the three-dimensional system (2) in a special but very interesting case, namely when the parameter R in (2c) is very small. This will be discussed and investigated in the next Section in detail.