The reproduction symbol is used to specify the rate of creation of new instances of a population submodel by each existing instance. It thus differs from the migration symbol, which specifies the total rate of creation of new instances.
The reproduction symbol only has meaning in the context of a population submodel. Therefore, it makes sense to construct a population submodel first, then add the reproduction symbol to it. This is not strictly necessary: you can add the reproduction symbol first, then construct a submodel around it, then make the submodel into a population submodel, but it is better practice to construct the population submodel first.
So, assuming that you already have a population submodel in your model diagram:
- The reproduction symbol can receive influence arrows from anywhere in the model, including from within the same submodel. This information can then be used to calculate the current value for the rate of creation of new instances of the population. As with all other relationships within a population submodel (except for the migration symbol), influences coming from within the same submodel relate to the properties of the same instance. So, for example, an influence from a variable “bodyweight” to the reproduction symbol indicates that it is each individual’s body weight that influences its own rate of reproduction.
- You can include as many reproduction symbols as you like. Biologically it would normally be pretty inappropriate to have more than one method for reproduction, but it is perfectly legal as far as Simile is concerned.
- Although it is possible to draw an influence arrow from the reproduction symbol to another element, the only use of this is with the channel_is( ) function. The value of the reproduction symbol is not the value entered or calculated, but the fraction remaining before the next instance is created. (See “Interpretation” below for details.)
The reproduction symbol captures the concept that, in many biological situations, the production of new individuals by those already in the population – reproduction – is an important mechanism for increasing population size. Moreover, the ability of an individual to reproduce will depend on its own characteristics: its age or weight, for example.
As with the migration symbol, Simile needs to resolve the fact that the value for reproduction is a floating point number, while new individuals can only be created one-by-one. The method it uses to do this is similar to that used for migration, and essentially involves the use of the reproduction term to contribute fractions of an individual to an accumulator: when the accumulator exceeds a whole number, then that number of new instances for the submodel are created, and the accumulator is reduced by the number of instances created.
There is, however, an important issue that the designers of Simile had to address. Should there be one accumulator for the whole population, or one for each of the current set of instances? In the former case, if you had five instances, each with a reproduction value of 0.1, then one new individual would be created every 2 time units. In the latter case, for the same settings, you would get no new instances for ten time units, then five would be created at the same time. The first approach seems more attractive, but suffers from a fatal flaw: it assumes that the parentage of newly-created individuals is irrelevant. This severely restricts the modelling you can do: in particular, it rules out modelling evolution, since that requires some concept of (biological) inheritance, which in turn means that each individual needs to know who its parent(s) are. Also, the second approach gives the same behaviour as the first if the value for reproduction is treated stochastically (e.g. as a probability), rather than as a precise deterministic contribution until you have enough credit to make one individual. Therefore, in Simile, each individual accumulates its own credit until it has sufficient to make one new individual.
The following cases illustrate how Simile interprets the information provided in the reproduction symbol. In all cases, the analysis is for ONE INDIVIDUAL in the current population: the total input of new individuals into the population is the sum of the input from each individual.
Three new instances are created every time step.
During the first time step, the value for reproduction (0.7) is not enough to create a new instance of the population submodel. Simile then remembers the value 0.7. When it comes to the next time step, the value for reproduction is then added on, giving 0.7+0.7=1.4. This is sufficient to create one new instance, leaving 0.4 in credit. The process continues, as shown here:
1 0.0 0.7 0.7 0 0.7
2 0.7 0.7 1.4 1 0.4
3 0.4 0.7 1.1 1 0.1
4 0.1 0.7 0.8 0 0.8
5 0.8 0.7 1.5 1 0.5
6 0.5 0.7 1.2 1 0.2
7 0.2 0.7 0.9 0 0.9
8 0.9 0.7 1.6 1 0.6
9 0.6 0.7 1.3 1 0.3
10 0.3 0.7 1.0 1 0.0
The value for reproduction is a reproduction rate, expressed in whatever your global unit of time is (typically “year”). Therefore a value of reproduction of 0.7 individuals per year gives a value of 0.07 individuals in one time step (0.1 year). Exactly the same procedure as above is applied, with the reproduction value accumulating until a whole number of individuals can be created.
The same procedure as above is applied, with three new instances being created in some time steps, and four in others.
You might well feel uncomfortable with the deterministic nature of the process. In this case, it is up to you to engineer a suitable stochastic mechanism that generates new individuals at the same average rate. In the future, Simile will include functions for sampling from appropriate frequency distributions for handling a variety of options (please contact the authors of Simile if you want to discuss this). In the meantime, and if your rate of reproduction is less than 1, then you can treat the value as a probability that reproduction will occur, and use the following expression for reproduction:
if rand_var(0,1)<0.7 then 1 else 0
where 0.7 is the reproduction rate.
This may be a pretty reasonable way of representing things, if it is the case that some individuals produce 1 offspring, and others produce none, with a probability of 0.7 and 0.3 respectively. (Biologically, it is possible that some individuals may produce 2 or more offspring, with the average being 0.7. In that case, the above method is only a rough approximation, and you should aim to develop a solution based on the relative probability of different numbers of offspring.)
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