Assume closed population (no I or E). NB: why? Because it makes it easier.

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1 What makes populations get larger? Birth and Immigration. What makes populations get smaller? Death and Emigration. B: The answer to the above?s are never things like "lots of resources" or "detrimental conditions." Those things only work as answers if they operate via altering birth, death, immigration and/or emigration rates. Is something ecology if it does not relate to birth rates, death rates, immigration rates, or emigration rates? o. Simple model of population growth (derivation of Exponential Growth Model): t+1 = t + B D + I E; where B, D, I, and E are in units of individuals per unit time (population-level rates). t+1 = t + B D t+1 - t = B D Δ t = B D. d/dt = B D. Assume closed population (no I or E). B: why? Because it makes it easier. Move t to left. B: why? To make the equation predict changes in size rather than size itself. Which equals... Transform into language of calculus. B: hidden assumption growth is instantaneous and continuous. Make B and D per capita rates. B: why? Because per capita rates don't depend on population size. B = b; where b is the per capita birth rate (ind/ind/time) and is population size. D = d; where d is the per capita death rate (ind/ind/time). d/dt = b d. d/dt = (b-d). d/dt = r. Extract the. b-d equals r, the intrinsic (instantaneous) population rate of increase; units are ind/ind/time. Integration (calculus stuff) leads to equation to predict in future. t = e rt ; where =initial population size; and e = base of natural logarithm. Example calculations: If r =.2, = 1, and we want to know the 5, then 5 = 1*(2.718 (.2)(5) ) = If =, and there are 5 births and 2 deaths during the next time step, then b = 5/ =.5 ind/ind/time; d = 2/ =.2 ind/ind/time; and r =.5-.2 =.3 ind/ind/time.

2 Descriptive powers of the model (when b>d, so that r>): d/dt d/dt (1/) time Assumptions of the model: 1. b and d are constant, so that r is constant. -if b>d, then population grows w/o bound; resources unlimited. 2. Population closed: no I or E. 3. o age or stage structure to population. -no differences in b or different ages or stages. 4. o genetic structure to population. -no fitness differences among individuals. 5. Growth is continuous with no time lags. -b and d happen instantaneously. -essentially, everything is born, gives birth, and dies at exactly the same time. Equilibrium states: solve for when d/dt =. r=; biologically unrealistic; statistically impossible. =; extinct or extirpated. Logistic population growth model: Incorporates density-dependent growth = d-d birth and death rates = intraspecific competition for resources = resource limitation. Make observed birth rates decline linearly with density (): b' = b a; where 'a' is the absolute slope of the descending line. Make observed death rates increase linearly with : d' = d + c; where 'c' is the slope of the ascending line. b d K

3 Substitution into exponential model and rearrangement leads to: d/dt = r(1-/k); which is equivalent to: d/dt = r([k-]/k); in both of which: K = (b-d) / (a+c); thus, the carrying capacity (K) is determined by intrinsic growth rate (b-d) divided by the effect of density dependence (a+c). K is a stable equilibrium: solve equation for when d/dt =. = r(1-/k) = 1-/K /K = 1 = K. Or = r([k-]/k) = (K-)/K = K- = K. Descriptive powers of the model (when b>d, so that r> and (b-d)>(a+c), so that K>1.): K d/dt d/dt (1/) time K K Population-level growth is fastest at = K/2. Integration of equation leads to prediction of population size in future: t = K / 1+ [(K- )/ ]e -rt. Assumptions of the model: 1. Remaining 4 from exponential model: -no age/stage structure. -no genetic structure. -no I or E. -growth is instantaneous and continuous. 2. Linear density-dependence. -observed b and d decline or increase in a linear fashion with changes in. 3. Constant carrying capacity. -K determined by b, d, a, and c; once determined, K is always the same. Example calculations: If b = 5, d = 2, a =.1, and c =.1, then K = (5-2) / (.1+.1) = 15. If you doubled the a & c, then K = 75.

4 Discrete Exponential and Logistic Growth Model: To deal with assumption of instantaneous growth, make models discrete (occur in absolute units of time). Introduces natural time-lag in action of d-d of 1 time unit, i.e., d-d at time=t affects growth at time=t+1 (no instantaneous growth). λ = 1 + r d ; where r d is the discrete growth factor (fractional change in from t to t+1). λ is the ratio of t+1 to t. e r = λ r = ln(λ) Discrete exponential: t+1 = λ t. Discrete logistic: To predict future : t = λ t. t+1 = t + r d t (1- t /K). B: can produce normal s-shaped rise to K; damped oscillations around K; 2-point, 4-point, and 8-point cycles; and chaos. Population Growth of 2 Competing Species: Back to instantaneous logistic growth models modified to include negative effect of presence of competing species. ew terms: ew models: α = per capita effect of Species 2 on the growth of Species 1. β = per capita effect of Species 1 on the growth of Species 2. d 1 /dt = r 1 1 (1-1 /K 1 -α 2 /K 1 ) d 2 /dt = r 2 2 (1-2 /K 2 -β 1 /K 2 ) B: α and β can be thought of as conversion factors from 1 species to the other (see the equations above to prove this to yourself in terms of their effects on per capita growth rates); if α=, then species 2 has no effect on species 1 (α 2 reduces to zero and the equation reduces to the normal 1-species logistic equation; same is true of β; if α>, then species 2 has some negative effect on species 1 population growth; if <α<1, then this effect is LESS THA the per capita effect of species 1 on itself, i.e., interspecific competition is weaker than intraspecific competition; if α>1, then this effect is GREATER THA the per capita effect of species 1 on itself, i.e., interspecific competition is stronger than intraspecific competition; all of the above applies to β, as well. α and β do not have to be related to each other mathematically: both could be >1, both could be <1, one could be high and the other low.

5 Solve for equilibrium: For Species 1 (set d 1 /dt = and solve for 1 ): 1 = K 1 - α 2 so, the equilibrium population size of 1 is equal to its own carrying capacity (K 1 ) minus some effect (α) of the presence of species 2 ( 2 ). If 1 = K 1 - α 2, then what is... 1 when 2 = K 1. 1 when α 2 = K 1. what is 2 at this point? K 1 /α. A state-space graph of 2 versus 1 for Species 1: K 1 /α 2 For Species 2 (same process leads to): 1 K 1 The x-axis is the answer the 1 st question above: the point plotted is (K 1,). The y-axis is the answer the 2 nd question above: the point plotted is (,K 1 /α). The line between them is the ZGI for Species 1, i.e., the combination of 1 and 2 at which d 1 /dt =, i.e., the possible equilibrium population sizes. If the system is to the right of this line, 1 will decrease to the line; if the system is to the left of this line, 1 will increase to the line (arrows). 2 = K 2 β 1 so, the equilibrium population size of 2 is equal to its own carrying capacity (K 2 ) minus some effect (β) of the presence of species 1 ( 1 ). If 2 = K 2 β 1, then what is... 2 when 1 = K 2. 2 when β 1 = K 2. what is 1 at this point? K 2 /β. A state-space graph of 2 versus 1 for Species 2: K K 2 /β The y-axis is the answer the 1 st question above: the point plotted is (,K 2 ). The x-axis is the answer the 2 nd question above: the point plotted is (K 2 /β,).

6 The line between them is the ZGI for Species 2, i.e., the combination of 1 and 2 at which d 2 /dt =, i.e., the possible equilibrium population sizes. If the system is above this line, 2 will decrease to the line; if the system is below this line, 2 will increase to the line (arrows). We put these graphs together to predict the outcome of competition between the species: Outcome 1 (Species 1 wins): K 1 = K 2 =. α =.5; β = 2. Outcome 3 (Winner depends): K 1 = K 2 =. α = 2.; β = /a /a 5 2 /B /B Outcome 2 (Species 2 wins): K 1 = K 2 =. α = 2.; β =.5 Outcome 4 (Stable Coexistence): K 1 = K 2 =. α =.5; β = /a /a 2 5 /B /B B: We are assuming equal carrying capacities to simplify and make the results depend entirely on α and β: when α is small (<1) and β is large (>1), species 1 wins; when reversed, species 2 wins; when both are large (>1), winner depends on initial combination of population sizes; when both are small (<1), stable coexistence occurs at the double-equilibrium point where the lines cross. B: If the species have different carrying capacities, it complicates things, but mostly just by altering how "small" and how "large" α and β have to be to produce the 4 possible results above.

7 Life Table Analysis: To deal with assumption of no age or stage structure. B: we are going back to assuming everything else, i.e., back to the continuous (calculus based) exponential growth model, but with age-structure. Example given below: x S(x) l(x) b(x) l(x)b(x) l(x)b(x)x Σ= R = Σl(x)b(x) = 5. offspring. Generation time (G): G = Σl(x)b(x)x / Σl(x)b(x) = 2.5 / 5. = 4.1 time units (e.g., years). r (intrinsic pop. growth rate): r = ln(r ) / G = 1.69/4.1 =.39 ind/ind/time unit; λ would = 1.47; r d would =.47.