Topic+8.+Single-species+Population+Growth

Topic+8.+Single-species+Population+Growth - Topic 8 Single...

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Topic 8: Single Species Population Growth I. Population – again, defined as those individuals of a given species in some place. II. Exponential population growth. A. The simplest form of population growth: N = N 0 e rt This equation gives the graph, as pictured, where N or N(t) is the current population size (at t), N 0 is the initial population size (at t = 0), e is the base of natural logs, and r is the intrinsic rate of increase: r = (birth rate – death rate)
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B. The differential equation of exponential population growth is easier to write: rN dt dN = The solution to this equation is the first equation given. Note that dN/dt is the slope of the graph, and is proportional to N: C. A population can’t grow exponentially forever. Would eventually be expanding from the earth at the speed of light and then couldn’t grow more. But it could show exponential, or near exponential growth for short periods of time: 1. Pheasant on Protection Island (Washington) Slide 2. Humans on earth. Slide Watt (emeritus professor at UCD) said it was “super-exponential”, e.g. 2 rN dt dN 2245
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III. Logistic population growth A. Puts an upper limit to population size called K (def), the carrying capacity – the population size (N) can’t get any bigger than K. B. The equation for the graph (solution to differential equation) is too hard to write for class purposes. Easier to write only the differential equation: - = κ N rN dt dN 1 Watch how the slope (dN/dt) varies. When N ≈ 0, like exponential (dashed line). If N = K, dN/dt = 0, rK (1 – 1) = 0. So slope approaches 0 at asymptote (which is K). As N increases, dN/dt first increases, then decreases. Maximum for logistic at N = K/2.
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C. The optimum yield is when dN/dt is maximum, for logistic at K/2. Population here is growing at fastest rate, and that is where it
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Topic+8.+Single-species+Population+Growth - Topic 8 Single...

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