2. Explanation of the model

Farmers produce some goods, and are assumed to grow logistically due to the limitation of natural resources. They themselves are again a resource for the bandits, which is exploited by these. Moreover the farmers are also taxed by the soldiers. The bandits are hunted by the soldiers, in order to protect their incomes from the farmers, and if the bandits are caught, they are executed by the soldiers. The groth dynamics of the soldiers is therefore governed both by the number of farmers to be protected, and by the number of bandits, that have to be hunted.
Both, the bandits and the soldiers, exhibit natural mortality or retirement.

Denoting by X(t), Y(t) and Z(t) the number of farmers, bandits, and soldiers, respectively, the dynamics of the system is described by the following system of ordinary differential equations:

where all the parameters are positive.

It can easily be checked, that if X(0), Y(0),Z(0)>=0 then X(t), Y(t),Z(t)>=0 for all t>=0, which means the system described by the above formulas is positive (as we would expect it to be, for the number of farmers, bandits and soldiers can't be negative).

The equation for X(t) gives the dynamics for the farmers.
In the absence of bandits and soldiers (that is for Y=Z=0) the farmers have a logistic growth, given by the term rX(1-(X/k)).
-aXY/(b+X) gives the mortality rate due to the predation by the bandits. This adopted form reflects the limited capability of the bandits in predating and processing the booty, and it is well known among ecologists [Hol65].
-hXZ is the mortality rate of the farmers due to subtraction of parts of their resources by the soldiers via taxation.

The equation for Y(t) gives the dynamics for the bandits.
In this equation the predation term aXY/(b+X) is transformed into a growth rate by multiplying it with a factor e.
The term -mY represents the natural mortality rate of the bandits.
The last term, -cYZ/(d+Y), is a mortality rate due to the action of the soldiers. Again, its form reflects the limited capability.

The equation for Z(t) finally gives the dynamics for the soldiers. It models the mechanism adopted in hiring soldiers.
It is assumed, that the hiring rate is proportional to the rate of criminality existing in the system at time t. This is expressed by the factor f to the rate of predation aXY/(b+X) of the farmers by the bandits.
The term -gZ is a rate of soldiers naturally leaving their job, either by retirement, or if they die.
Notice that the bandits are not considered as a cause of mortality for the soldiers,since it is assumed, that training and equippment of the soldiers are much better, than that of the bandits, and therefore the probability of dying in a fight is so small, that it can be neglected here.

It is interesting, that the "dynamic cycle" system described above has relevant analogies with what in ecology is called a food-chain. A food-chain is a system, describing the interactions among vegetal and/or animal species where each species gets its resources by predation of lower level species, and in its turn, might be a resource for higher level species (e.g.[May73]).
Compared to these systems, the farmers, bandits and soldiers can be considered as, respectively, preys, predators, and superpredators. The soldiers even exploit both of the other species. The main pecularity in this dynastic cycle model with respect to the standard food-chains (e.g. [HaPo91]) lies in the form of the equation for Z(t), which , as already has been pointed out above, relates the hiring rate of new soldiers (fixed by a public authority) to the criminality rate in the whole system, represented by the rate of predation of the farmers by bandits.

In the next Section we will simplify the model by rescaling.