Mathematics 22: Lecture 4

Transcription

1 Mathematics 22: Lecture 4 Population Models Dan Sloughter Furman University January 10, 2008 Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

2 Malthusian growth model Let p(t) denote the population of some animal species in some specified environment at t. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

3 Malthusian growth model Let p(t) denote the population of some animal species in some specified environment at t. ṗ Thomas Malthus proposed the model: = r for some constant p r > 0. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

4 Malthusian growth model Let p(t) denote the population of some animal species in some specified environment at t. ṗ Thomas Malthus proposed the model: = r for some constant p r > 0. Note: r is the per capita growth rate. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

5 Malthusian growth model Let p(t) denote the population of some animal species in some specified environment at t. ṗ Thomas Malthus proposed the model: = r for some constant p r > 0. Note: r is the per capita growth rate. Solution: Suppose the initial condition is p(0) = p 0. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

6 Malthusian growth model Let p(t) denote the population of some animal species in some specified environment at t. ṗ Thomas Malthus proposed the model: = r for some constant p r > 0. Note: r is the per capita growth rate. Solution: Suppose the initial condition is p(0) = p 0. Then t 0 1 t p(s) p (s)ds = rds. 0 Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

7 Malthusian growth model Let p(t) denote the population of some animal species in some specified environment at t. ṗ Thomas Malthus proposed the model: = r for some constant p r > 0. Note: r is the per capita growth rate. Solution: Suppose the initial condition is p(0) = p 0. Then t And so log(p(t)) log(p 0 ) = rt. 0 1 t p(s) p (s)ds = rds. 0 Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

8 Malthusian growth model Let p(t) denote the population of some animal species in some specified environment at t. ṗ Thomas Malthus proposed the model: = r for some constant p r > 0. Note: r is the per capita growth rate. Solution: Suppose the initial condition is p(0) = p 0. Then t And so log(p(t)) log(p 0 ) = rt. Solving for p: p = p 0 e rt. 0 1 t p(s) p (s)ds = rds. 0 Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

9 Malthusian growth model Let p(t) denote the population of some animal species in some specified environment at t. ṗ Thomas Malthus proposed the model: = r for some constant p r > 0. Note: r is the per capita growth rate. Solution: Suppose the initial condition is p(0) = p 0. Then t And so log(p(t)) log(p 0 ) = rt. Solving for p: p = p 0 e rt. 0 1 t p(s) p (s)ds = rds. 0 Hence the Malthusian model leads to exponential population growth. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

10 Malthusian growth model Let p(t) denote the population of some animal species in some specified environment at t. ṗ Thomas Malthus proposed the model: = r for some constant p r > 0. Note: r is the per capita growth rate. Solution: Suppose the initial condition is p(0) = p 0. Then t And so log(p(t)) log(p 0 ) = rt. Solving for p: p = p 0 e rt. 0 1 t p(s) p (s)ds = rds. 0 Hence the Malthusian model leads to exponential population growth. Can that be realistic? Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

11 The logistics model We should expect that as the population grows, and resources become more scarce, r will decrease. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

12 The logistics model We should expect that as the population grows, and resources become more scarce, r will decrease. Simple idea: Replace r with r ( 1 p K ) for some constant K > 0. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

13 The logistics model We should expect that as the population grows, and resources become more scarce, r will decrease. Simple idea: Replace r with r ( 1 p ) K for some constant K > 0. ( Logistics model: ṗ = r 1 p ) p = rp r K K p2. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

14 The logistics model We should expect that as the population grows, and resources become more scarce, r will decrease. Simple idea: Replace r with r ( 1 p ) K for some constant K > 0. ( Logistics model: ṗ = r 1 p ) p = rp r K K p2. Note: p(t) = 0 and p(t) = K, for all t, are equilibrium solutions. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

15 The logistics model We should expect that as the population grows, and resources become more scarce, r will decrease. Simple idea: Replace r with r ( 1 p ) K for some constant K > 0. ( Logistics model: ṗ = r 1 p ) p = rp r K K p2. Note: p(t) = 0 and p(t) = K, for all t, are equilibrium solutions. Moreover, ṗ > 0 when 0 < p < K, and ṗ < 0 when p > K. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

16 The logistics model We should expect that as the population grows, and resources become more scarce, r will decrease. Simple idea: Replace r with r ( 1 p ) K for some constant K > 0. ( Logistics model: ṗ = r 1 p ) p = rp r K K p2. Note: p(t) = 0 and p(t) = K, for all t, are equilibrium solutions. Moreover, ṗ > 0 when 0 < p < K, and ṗ < 0 when p > K. See the phase line plot on page 31: p = K is an attractor, whereas p = 0 is a repeller. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

17 Autonomous equations Consider an autonomous equation du dt = f (u). Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

18 Autonomous equations Consider an autonomous equation du dt = f (u). If f (u ) = 0, then u(t) = u is a solution for all t. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

19 Autonomous equations Consider an autonomous equation du dt = f (u). If f (u ) = 0, then u(t) = u is a solution for all t. The solution is an attractor, or a stable equilibrium, if f (u) > 0 to the left of u and f (u) < 0 to the right of u. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

20 Autonomous equations Consider an autonomous equation du dt = f (u). If f (u ) = 0, then u(t) = u is a solution for all t. The solution is an attractor, or a stable equilibrium, if f (u) > 0 to the left of u and f (u) < 0 to the right of u. The solution is an repeller, or a unstable equilibrium, if f (u) < 0 to the left of u and f (u) > 0 to the right of u. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

21 Autonomous equations Consider an autonomous equation du dt = f (u). If f (u ) = 0, then u(t) = u is a solution for all t. The solution is an attractor, or a stable equilibrium, if f (u) > 0 to the left of u and f (u) < 0 to the right of u. The solution is an repeller, or a unstable equilibrium, if f (u) < 0 to the left of u and f (u) > 0 to the right of u. Otherwise, the solution is semi-stable Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

22 Example Consider du dt = u2 (4 u 2 ). Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

23 Example Consider du dt = u2 (4 u 2 ). The equilibrium solutions are u(t) = 2, u(t) = 0, and u(t) = 2. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

24 Example Consider du dt = u2 (4 u 2 ). The equilibrium solutions are u(t) = 2, u(t) = 0, and u(t) = 2. u(t) = 2 is an unstable equilibrium, u(t) = 0 is a semi-stable equilibrium, and u(t) = 2 is a stable equilibrium. Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6

25 Example (cont d) Direction field for du dt = u2 (4 u 2 ): Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, / 6