Population Growth Chapter 11 Growth With Discrete Generations Species with a single annual breeding season and a life span of one year ex annual plants Population growth can then be described by the following equation Nt 1 R0Nt Where Nt population size of females at generation t Nt 1 population size of females at generation t 1 R0 net reproductive rate or number of female offspring produced per female generation Population growth is very dependent on R0 Multiplication Rate R0 Constant If R0 1 the population increases geometrically without limit If R0 1 then the population decreases to extinction The greater R0 is the faster the population Geometric Growth increases Multiplication Rate R0 Dependent on Population Size Carrying Capacity the maximum population size that a particular environment is able to maintain for a given period At population sizes greater than the carrying capacity the population decreases At population sizes less than the carrying capacity the population increases At population sizes the carrying capacity the population is stable Equilibrium Point the population density that the carrying capacity Net Reproductive rate R0 as a function of population density Y mX b Y b m X N 100 then R0 1 0 population stable N 100 then R0 1 0 population decreases Intercept N 100 then R0 1 0 population increases Remember at R0 1 0 birth rates death rates We can measure population size in terms of deviation from the equilibrium density z N Neq Where z deviation from equilibrium density N observed population size Neq equilibrium population size R0 1 0 R0 1 0 B N Neq When N Neq then R0 1 0 Where R0 net reproductive rate y intercept b will always 1 0 population is stable B slope of line m the B comes from a regression coefficient With these equations z N Neq R0 1 0 B N Neq We can substitute R0 in Nt 1 R0Nt to get Nt 1 1 0 B zt Nt How much the population will change R0 Start with an initial population Nt of 10 a slope B 0 009 and Neq 100 and the population gradually reaches 100 and stays there Nt 1 1 0 B z Nt 1 10 00 2 18 10 3 31 44 4 50 84 5 73 34 6 90 93 7 98 35 8 99 81 9 99 98 10 100 00 11 100 00 12 100 00 The population reaches stabilization with a smooth approach 1 10 00 2 26 20 3 61 00 4 103 82 5 96 68 6 102 46 7 97 92 8 101 58 9 98 69 10 101 02 11 99 17 12 100 65 13 99 47 14 100 42 15 99 66 16 100 27 17 99 78 18 100 17 19 99 86 20 100 11 Start with an initial population Nt of 10 a slope B 0 018 and Neq 100 and the population oscillates a little bit but eventually 64 generations stabilizes at 100 and stays there This is called convergent oscillation Nt 1 1 0 B z Nt 1 10 00 2 32 50 3 87 34 4 114 98 5 71 92 6 122 41 7 53 84 8 115 97 9 69 67 10 122 50 11 53 60 12 115 78 13 70 11 14 122 50 15 53 59 16 115 77 17 70 12 18 122 50 19 53 59 20 115 77 Start with an initial population Nt of 10 a slope B 0 025 and Neq 100 and the population oscillates with a stable limit cycle that continues indefinitely Nt 1 1 0 B z Nt 1 10 00 2 36 10 3 103 00 4 94 05 5 110 29 6 77 39 7 128 13 8 23 59 9 75 87 10 128 96 11 20 64 12 68 14 13 131 10 14 12 87 15 45 40 16 117 28 17 58 51 18 128 91 19 20 84 20 68 68 Start with an initial population Nt of 10 a slope B 0 029 and Neq 100 and the population fluctuates chaotically Nt 1 1 0 B z Nt B Population 0 009 Gradually approaches equilibrium 0 018 Convergent oscillation 0 025 Stable limit cycles 0 029 Chaotic fluctuation As the slope increases the population fluctuates more A high B causes an overshoot towards stabilization Remember B is the slope of the line and represents how much Y changes for each change in X Define L as B Neq The response of the population at equilibrium L between 0 and 1 Population approaches equilibrium without oscillations L between 1and 2 Population undergoes convergent oscillations L between 2 and 2 57 Population exhibits stable limit cycles L above 2 57 Population fluctuates chaotically Growth With Overlapping Generations Previous examples were for species that live for a year reproduce then die For populations that have a continuous breeding season or prolonged reproductive period we can describe population growth more easily with differential equations Multiplication Rate Constant In a given population suppose the probability of reproducing b is equal to the probability of dying d r b d Nt Then rN b d N N ert 0 Where Nt population at time t t time r per capita rate of population growth b instantaneous birth rate d instantaneous death rate Population grows geometrically Nt 1 R0Nt We can determine how long it will take for a population to double Nt rt 2 e N0 Loge 2 rt Loge 2 r t r realized rate of population growth per capita For example r t 0 01 69 3 0 02 34 7 0 03 23 1 0 04 17 3 0 05 13 9 0 06 11 6 Multiplication Rate Dependent on Population Size dN K N rN dt K Where N population size t time r intrinsic capacity for increase K maximal value of N carrying capacity K r Pop Size K N K Growth Rate 1 1 99 100 0 99 1 25 25 100 6 25 1 50 50 100 25 1 75 25 100 18 75 1 95 5 100 4 75 1 99 1 100 0 99 1 100 0 100 0 Logistic population growth has been demonstrated in the lab Year to year environmental fluctuations are one reason that population growth can not be described by the simple logistic equation Time Lag Models Animals and plants do not respond immediately to environmental conditions Change our assumptions so that a population responds to t 1 population size not the t population size L Bneq If 0 L 0 25 then stable equilibrium with no oscillation If 0 25 L 1 0 then convergent oscillation If L 1 0 then stable limit cycles or divergent oscillation to extinction Ex Daphnia Stochastic Models Models discussed so far are deterministic given certain conditions each model predicts one exact condition However biological systems are probabilistic what is the probability that a female will have a litter in the next unit of time What is the probability that a female will have a litter of three instead of four Natural population trends are …