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Lecture

# lecture note 20 for BGYB50

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University of Toronto Scarborough

Biological Sciences

BIOC50H3

Herbert Kronzucker

Winter

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LECTURE 20: - Competition affects per capita survivorship, susceptibility to disease, and, very predictably, body size and weight (Yoda’s Law: individual body weight and density are correlated inversely; in a log-log plot of these parameters, a linear relationship is frequently found, with a slope of -3/2; example: forest stands: trees are smaller when there are many of them compared to only a few; conversely, if trees “want” to grow larger, they will have to reduce their own density through self-thinning) - In order to describe the competitive interactions between populations of two species, we need to modify the logistic equations for those populations by incorporating the per capita competitive effect (!) each has upon the other - ! gives the relative “muscle power” of the two species populations in relation to each other - Subscripts to ! denote the directionality of the competition effect (!12 = the per capita competitive effect on species 1 of species 2; !21 = the per capita competitive effect on species 2 of species 1); e.g. !12 = 0.5, if the average individual in the population of species 2 is half “as strong” as the average individual in the population of species 1; in other words, multiplying N2 by !12 converts any given number of individuals of population 2 to a number of “N1 equivalents” - The logistic equations then become: www.notesolution.com
- These are known as the Lotka-Volterra equations for competition. While the original logistic equations adequately describe the exponential growth phase and the deflection from exponentiality at carrying capacity (which is due to environmental resistance and intraspecific competition), the modified equations consider the additional “slow-down” effect brought about by an interfering population (usu. of another species) competing for the same “space” (well, any limiting resource) - To describe the properties of the Lotka-Volterra model, we must ask: When does the number of individuals of species 1 increase and when does it stop to increase? And, what are the analogous circumstances for species 2? - Clearly, populations 1 and 2 will stop to grow when dN1/dt = 0, and dN2/dt = 0, respectively - This condition will be fulfilled when r = 0 or N = 0, or when, less trivially, the numerator K1 – N1 - !12N2 = 0; in the latter case, our equation (for species 1) can be rewritten to: N1 = K1 - !12N2, which now represents a simple linear equation (similar to: y = b + mx, where m is the slope of the line and b is the intercept with the y-axis), i.e. we can plot N1 vs. N2 and will obtain a straight line (called the isocline) whose y-intercept will be K1; if y = 0 (i.e. N1 = 0), we can get the x-intercept, which is then K1/!12 (K1/!12 can be seen as the “carrying capacity equivalency” for species 2 with respect to species 1, or the number of individuals of species 2 that would be needed to competitively match the number of individuals of species 1 when the latter is at its carrying capacity) - The same can be done for species 2, and both lines (isoclines) and intercepts can be graphed in the same plot (which, by convention, has N1 as the x-axis and N2 as the y-axis) www.notesolution.com
- The isoclines in these graphs delineate the areas where the numbers of populations 1 and 2 can increase (to the left of the respective isoclines) or have to decrease (to the right of the isoclines) - Four possible scenarios can now be envisaged (you must read your textbook on this to not be completely baffled!), depending on the relative positioning of K1, K2, K1/!12, and K2/!21 on the axes (and, consequently, on whether the two isoclines cross each other’s paths or run independently): each has a different outcome of the competition www.notesolution.comLECTURE 20: - Competition affects per capita survivorship, susceptibility to disease, and, very predictably, body size and weight (Yodas Law: individual body weight and density are correlated inversely; in a log-log plot of these parameters, a linear relationship is frequently found, with a slope of -32; example: forest stands: trees are smaller when there are many of them compared to only a few; conversely, if trees want to grow larger, they will have to reduce their own density through self-thinning) - In order to describe the competitive interactions between populations of two species, we need to modify the logistic equations for those populations by incorporating the per capita competitive effect () each has upon the

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