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${\tfrac {dN}{dt}}=rN(1-{\tfrac {N}{K}})$

5(c)

$N$ represents the number of individuals in the population, $r$ represents the relative rate at which the members of the population reproduce when unchecked, and $K$ represents the carrying capacity of the environment. Though quite simple, the logistic equation displays quite interesting behavior across a wide spectrum of circumstances. When $N$ is low—when there are relatively few members of a population—growth can proceed almost unchecked, as the first term on the right side of the equation dominates. As the population grows in size, though, the value of ${\tfrac {N}{K}}$ increases, making the carrying capacity of the environment—how many (say) deer the woods can support before they begin to eat themselves out of house and home—becomes increasingly important. Eventually, the contribution of ${\tfrac {N}{K}}$ outpaces the contribution of $rN$, putting a check on population growth. More sophisticated versions of the logistic equation—versions in which, for instance, $K$ itself varies as a function of time or even as a function of $N$—show even stronger non-linear behavior.^[1] It is this interrelationship between the variables in the equation that makes models like this one non-linear. Just as with the Olympian treadmill we described above, the values of the relevant variables in the system of differential equations describing the system depend on one another in non-trivial ways; in the case of the treadmill, the value of a button-press varies with (and affects) the speed of the belt, and in the case of the logistic equation, the rate of population growth varies with (and affects) extant population. This general

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1. Consider, for instance, a circumstance in which the carrying capacity of an environment is partially a function of how much food is present in that environment, and in which the quantity of food available is a function of the present population of another species. This is often the case in predator-prey models; the number of wolves an environment can support partially depends on how many deer are around, and the size of the deer population depends both on how much vegetation is available for the deer to eat and on how likely an individual deer is to encounter a hungry wolf while foraging.

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