,$ x_{n+1} = r x_{n}(1 - x_{n})$
where x represents the ratio of the existing population to the maximum possible population, n represents the number of iterations, and r is a parameter ranging between 0 and 4 that changes the behaviour of the population growth.
In this simulation (link here), we denote the parameter r by 4r, where r now ranges between 0 and 1. From here on, when we refer to r, we mean the r in the simulation that ranges between 0 and 1.
This model simulates the effects of reproduction, where the population size increases at a rate proportional to the current population when the population is small, and starvation, where the population size decreases at a rate proportional to the carrying capacity of the environment.
It is not decisive for the bifurcation that the limiting term is exactly (1-x(n)). The crucial point is the non-linearity of the conjunction x(n) -x(n)^2.
To demonstrate this, a generalized series rule is used in this simulation, using a term (1-x(n)^k), with k > 0 :
x(n+1) = 4rx(n)(1-x(n)^k)
When r is approximately 0.89 to 1, the population size starts to exhibit chaotic behaviour, fluctuating wildly between many values. In fact, the Logistic Map is commonly used to show how chaotic systems can arise from seemingly simple models.
where x represents the ratio of the existing population to the maximum possible population, n represents the number of iterations, and r is a parameter ranging between 0 and 4 that changes the behaviour of the population growth.
In this simulation (link here), we denote the parameter r by 4r, where r now ranges between 0 and 1. From here on, when we refer to r, we mean the r in the simulation that ranges between 0 and 1.
This model simulates the effects of reproduction, where the population size increases at a rate proportional to the current population when the population is small, and starvation, where the population size decreases at a rate proportional to the carrying capacity of the environment.
It is not decisive for the bifurcation that the limiting term is exactly (1-x(n)). The crucial point is the non-linearity of the conjunction x(n) -x(n)^2.
To demonstrate this, a generalized series rule is used in this simulation, using a term (1-x(n)^k), with k > 0 :
x(n+1) = 4rx(n)(1-x(n)^k)
When r is approximately 0.89 to 1, the population size starts to exhibit chaotic behaviour, fluctuating wildly between many values. In fact, the Logistic Map is commonly used to show how chaotic systems can arise from seemingly simple models.
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| Logistic Map when k = 1 |
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| Logistic Map when k = 1.5 |
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