Population Growth: Exponential Growth and Matrix Model - Prof. Eric M. Schauber, Study notes of Zoology

Download Population Growth: Exponential Growth and Matrix Model - Prof. Eric M. Schauber and more Study notes Zoology in PDF only on Docsity! 1 Population Growth Population Dynamics changes in population size that result from variations in the rates of birth, death and movement of individuals Population Immigration + Births + Emigration - Deaths- • Why are rabbits the smartest animals in the world? • Answer: because the know how to multiply so well! An Old, Bad Joke POPULATION GROWTH IS EXPONENTIAL: A PROCESS OF REPEATED MULTIPLICATION • If per capita growth rate is constant, a population will grow exponentially. Nt+∆t = Nt*X∆t First Law of Population Growth • Discrete – Reproduction occurs in distinct pulses (usually annual) – Newborns cannot reproduce until next pulse • Continuous – Reproduction occurs throughout the year or growing season – Newborns can reproduce in the year they are born – “Compounding” occurs within each time step (e.g., year) Discrete vs. Continuous Population Growth Discrete vs. Continuous Population Growth 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 6 Time N 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 6 Time N 2 Discrete vs. Continuous Population Growth: Credit Cards Mr. Nice Guy Bank: Finance Charges (12% APR) Compounded Annually You owe $100 today, and don’t make any payments (assume no late fees!) In 1 year you will owe: $100 X 1.12 = $112 Discrete vs. Continuous Population Growth: Credit Cards Snitty Bank: Finance Charges (12% APR) Compounded Monthly (12%/12 months = 1%/month) You owe $100 today, and don’t make any payments (assume no late fees!) In 1 month you will owe: $100 X 1.01 = $101 In 2 months you will owe: $100 X 1.01 X 1.01 = $102.01 In 1 year you will owe: $100 X (1.01)12 = $112.68 Discrete vs. Continuous Population Growth: Credit Cards Instantaneous Bank of America: Finance Charges (12%) Compounded Instantaneously You owe $100 today, and don’t make any payments (assume no late fees!) In 1 year you will owe: $100 X e0.12 = $112.75 e = 2.71828183… (root of the natural logarithm) Describing Population Growth 0 0.2 0.4 0.6 0.8 1 1.2 0 1 Time N N0 N0(1-d) N0d N1 N0(1-d)b N1 = N0(1-d)(1+b) λ = (1-d)(1+b) deaths births • Discrete λ = (1 - d)(1 + b) Total Growth Rate: Nt+1 – Nt = (1-λ)Nt Solution: Nt = N0*λt • Continuous r = b - d Total Growth Rate: dN/dt = rN Solution: Nt = N0*ert (same as discrete, but with λ = er) Either way, doubling time = log(2)/log(λ) Discrete vs. Continuous Exponential Population Growth Exponential Growth 0 100 200 300 400 500 0 5 10 Time (t ) Po pu la tio n Si ze ( N t) 0.5 0.8 1 1.2 1.5 λ 5 So, putting it all together, if there are 7 age classes (0-6): n0,t = n0,t-1m0 + n1,t-1m1 + n2,tm2 +n3,tm3 +n4,tm4 +n5,tm5 +n6,tm6 n1,t = n0,t-1s0 n2,t = n1,t-1s1 n3,t = n2,t-1s2 n4,t = n3,t-1s3 n5,t = n4,t-1s4 n6,t = n5,t-1s5 Nt = n0,t + n1,t + n2,t + n3,t + n4,t + n5,t + n6,t THAT’S A LOT OF WORK JUST TO WRITE! Maybe there’s an easier way... Population Growth With Age Structure Population Growth With Age Structure: MATRIX MODEL L × Nt = Nt+1 = m0 s0 0 0 0 0 0 m1 0 s1 0 0 0 0 m6 0 0 0 0 0 0 m5 0 0 0 0 0 s5 m4 0 0 0 0 s4 0 m3 0 0 0 s3 0 0 m2 0 0 s2 0 0 0 Leslie Projection Matrix n0,t n1,t n2,t n3,t n4,t n5,t n6,t n0,t+1 n1,t+1 n2,t+1 n3,t+1 n4,t+1 n5,t+1 n6,t+1 population vector Nt+1L Nt × Population Growth With Age Structure: MATRIX MODEL L × Nt = Nt+1 As this process is repeated, the population converges on a stable age distribution (SAD) as t gets large, nx,t/Nt approaches a constant value (different for each age class, but constant over time) growth is truly exponential after SAD is achieved 0 10 20 30 40 50 60 0 1 2 3 4 Age Class # Fe m al es 0 10 20 30 40 50 60 70 0 1 2 3 4 Age Class # Fe m al es0 1 2 3 4 Year t Year t+1 NOT at stable age distribution AT stable age distribution 0 10 20 30 40 50 60 70 0 1 2 3 4 Age Class # Fe m al es 0 1 2 3 4 0 1 2 3 4 Population Growth With Age Structure: MATRIX MODEL L × Nt = Nt+1 Why are you doing this to us, Evil Dr. Schauber? What’s the point of making these horrible matrices? Using linear algebra (“matrix math”), L can give us: •the stable age distribution (= dominant right eigenvector) •the asymptotic per capita growth rate (= dominant eigenvalue) •the reproductive value of each age class (= dominant left eigenvector) •(r.v. is the contribution of each individual in an age class to overall pop. growth rate) Population Growth With Age Structure: MATRIX MODEL How can matrix models be useful for management? 6 Things to Remember • Exponential growth: discrete vs. continuous time – both are linear on a logarithmic scale • Age structure: Matrix models – what are N and L? – what are their properties? – what is stable age distribution? – advantages of matrix approach – utility for management