In this section we will present a few examples of models of biological and ecological systems. Normally little exact knowledge is available about such systems, which would be needed for detailed physical modeling. This forces researchers to use inductive models based on experimental observations rather than physical intuition of system dynamics, which has the disadvantage that the models are difficult to validate for predictions outside the observed range.A combined approach is possible, where some approximate physical knowledge is combined with adapting model coefficients based on experimental observations. Such an approach will be explored in the subsequent models where we will investigate a continuous-time model for population dynamics and a discrete-time cyclic growth model for the energy forest.
In this example we will study the dynamics of populations of animals in an idealized ecological system.We assume that a population contains P individuals. In general, the change rate of the number of animals in a population is the difference between the population growth rate or birth rate and its death rate.The modelica class is presentend below
As expected, the plot shows an exponentially increasing population. We should, however, remember that exponential growth can never persist indefinitely in a closed system with a limited food supply
Now we would like to study a more interesting system consisting of two interacting populations of predators and prey animals. The simplest model of this kind is the so-called Lotka-Volterra model. In this case, the predators are foxes and the prey animals are rabbits. The rabbit population of size R is assumed to have an unlimited food supply. On the other hand, the fox population of size F feeds on the rabbits. The rate of rabbit deaths can be assumed to be proportional to the number of foxes due to increased hunting pressure, and to the number of rabbits due to the higher probability of success in hunting. The modelica class is presented below
The Lotka-Volterra model has a rather special property: the solution variables, i.e., the rabbit and fox population sizes, do not approach steady-state values as time goes by. Instead, they approach periodic steady-state values- the solution oscillates. The shape of the oscillation is very characteristic for Lotka-Volterra models and corresponds rather well to several experimental population studies of predators and prey animals. However, this does not prove our partly inductive model equations to be correct, only that they seem to have certain approximate physical significance.
Simplified approaches to World models can describe an interesting set of behaviors while keeping model complexity at a minimum. One such very simplified model is the Handy model, an acronym for Human and Nature Dynamical model.
In the Handy model, the total population is divided between two groups: Commoners and Elite, of population sizes commoners and elite respectively. The population grows through birth rates birthRComm and birthRElite, which are constants. The population shrinks through death rates deathRComm and deathRElite which are in turn defined as functions of wealth. The main dynamics part of the model is defined by the following four differential equations:
der(commoners) = birthRComm*commoners - deathRComm*commoners;
der(elite) = birthRElite*elite deathRElite*elite;
der(nature) = natureRegen*nature*(natureCap-nature) depletFactor*nature;
der(wealth) = depletFactor*commoners*nature consRComm - consRElite;
The modelica model of handybase is presented below
The Handy model can be used to define three types of societies:
Egalitarian society no elite, i.e., the elite population elite = 0. Scenario models HandyEgal1 to ModelEgal4.
Equitable society commoners and elite, both groups earn the same per person, i.e., ineqFactor=1. Scenario models HandyEquit1 to HandyEquit5.
Unequal society commoners and elite, an elite person earns more than a commoner; ineqFactor>=1. Scenario models HandyUnEq1 to HandyUnEq4.
For this type of society, the optimal value for the depletion factor that maximizes the carrying capacity (carryingCap) is depletFactorEq=6.67E-6, which has been derived analytically (Motesharrei, Rivas, and Kalnay 2013). Below is the HandyEgalitarianBase model which is inherited by all Egal models
Scenario Model HandyEgal1: Soft landing to Equilibrium
Scenario Model HandyEgal3: Cycles of Prosperity and Collapse . In this scenario we increase the depletion intensity by selecting depletFactor = 4 * depletFactorEq.
Scenario HandyEquit1: Soft Landing to Optimal Equilibrium
Scenario HandyEquit2: Oscillatory Approach to Equilibrium . If the depletion factor is further incremented by making depletFactor = 2.64 * depletFactorEq we obtain an oscillatory behavior.
The unequal society appears to be closer to the status of our current world. The inequality factor ineqFactor is made to range from 10 to 100 in the unequal scenarios.We will reproduce a pair of experiments intended to show the effect of birth control as a means for moving from an unsustainable into a sustainable mode of behavior.
Scenario HandyUneq2: Full Collapse . In this scenario the inequality factor is set very high: ineqFactor = 100. Also an initial seed for the population of the Elite is set to elite = 0.2 and a large depletion factor (including pollution effects) depletFactor = 1E-4 are selected
Scenario HandyUneq3: Soft Landing to Optimal Equilibrium . In this scenario several parameters and initial values are changed with respect to the previous case.
Inequality is reduced a factor ten, yielding ineqFactor = 10.
The depletion factor is set to its equilibrium value, derived analytically: depletFactor = depletFactorEq = 6.35E-6, much lower than the previous scenario.
The initial values for the population are set as commoners=10 000 and elite=3000.
The birth rates are assumed to be controllable and selected as birthRComm=0.065 and birthRElite=0.02