Lecture 3. Dynamical Systems in Continuous Time
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1 Lecture 3. Dynamical Systems in Continuous Time University of British Columbia, Vancouver Yue-Xian Li November 2,
2 3.1 Exponential growth and decay A Population With Generation Overlap Consider a population that satisfies the following. Different generations can overlap. Population size, N(t), varies continuously. Average per capita birth rate is: b (> 0). Average per capita death rate is: d (> 0). Let RoC, RoB, RoD denote rates of change, birth, and death respectively. Thus, RoC in N(t) = RoB - RoD, which leads to N (t) = bn(t) dn(t), where b and d are parameters. 2
3 The model can be simplified to N (t) = (b d)n(t) = rn(t), where parameter r = b d ( < r < ) is the net per capita growth rate. r > 0 if b > d, r < 0 if b < d. IC: N(t 0 ) = N 0, often t 0 = 0. This is a single-variable/1d differential equation. This is a dynamical system in continuous time. This is a linear, homogeneous differential equation. 3
4 Solving the differential equation dn dt = rn(t). Separation of variables: dn N = rdt dn N = rdt, yielding ln N = rt + C 0 N(t) = e rt+c 0 = e C 0e rt. Therefore, N(t) = Ce rt. 4
5 Applying the IC N(0) = N 0 : N 0 = N(0) = Ce 0 = C, C = N 0. Therefore, the solution is N(t) = N 0 e rt. If r > 0, N(t) grows exponentially. If r = 0, N(t) remains constant at N(t) = N 0. If r < 0, N(t) decays exponentially. 5
6 Graphing of the result Online graph plotter Figure 1: N(t) = e rt for r = 0.5 (blue), r = 0.5 (red), and r = 0 (green). 6
7 Graphing of the result in log scale Online graph plotter Figure 2: N(t) = e rt for r = 0.5 (blue), r = 0.5 (red), and r = 0 (green). 7
8 Comparison to experimental data Adopted from A. Hastings (1997) Population Biology. Concepts and Models. 8
9 Summary: A DE is typically an equation in which the derivative of an unknown function is related to itself and/or other quantities (which may include other derivatives of the function). It typically has infinitely many solutions that differ by one or more constants. If the value of the function at some initial time point (IC) is given, then the solution that satisfies both the DE and the IC is typically unique, i.e. the arbitrary constant in the general solution is typically determined uniquely by the IC. Solutions to a linear, homogeneous DE is typically an exponential function which either grows exponentially to infinity or decreases exponentially to zero. 9
10 3.2 Density-dependent growth: logistic equation Consider a single population with generation overlap. Net average per capita growth rate is density-dependent: r(n) = a(1 N K ) (a > 0, K >> 1). r(n) decreases as N increases. When N << K, r a=const. When N K, r is very small, representing reduced reproduction rate at high population density. 10
11 Equation of logistic model in continuous time The equation that governs the time evolution of the model variable (also called state variable) N (t) = r(n)n = an(1 N K ), N 0 = N(t = 0). This is a nonlinear differential equation. a, K > 0 are two parameters. 11
12 Nondimentionalized/dimensionless form Introduce a new, dimensionless variable n(t) = N(t) K, τ = rt where n(t) is the size of the population at t measured in units of K. Thus, dividing both sides of the equation by K 1 dn K dt = rn K (1 N K ) 1 dn r dt = n(1 n) dn dτ = n(1 n), which contains only one parameter a. Values of the original model parameters r, K do not change the dynamics of the system. General solution in closed form can be obtained. Unlike the logistic equation in discrete time, nothing chaotic can occur in this system. 12
13 Solving the differential equation dn dτ = n(n 1). Separation of variables: dn n(n 1) = dτ dn n(n 1) = dτ, yielding ln n 1 n = τ + C 0 n 1 n = e τ+c 0 = e C 0e τ = Ce τ. Therefore, n(t) = 1 1 Ce τ. 13
14 Applying the IC n 0 = N 0 K : n 0 = 1 1 C, C = n 0 1 n 0. Therefore, the solution is 1 n(τ) = = n n 0 n e τ n 0 +(1 n 0 )e τ. 0 In original variable, N(t) = KN 0 N 0 +(K N 0 )e rt. When τ = 0 (t = 0), n(0) = n 0 (N(0) = N 0 ). When τ (t ), n(t) 1 (N(t) K). Long term behaviour: n(τ) 1 (N(t) K) as τ, t irrespective of the value of n 0 ( 0). Question: Who can tell how do the solution curves look like based on the expression above? 14
15 Graphing of the result Online graph plotter Figure 3: n(t) = n 0 n 0 +(1 n 0 )e at for a = 1 and n 0 = 0.1 (blue), n 0 = 0.51 (red), and n 0 = 1.25 (green). 15
16 Comparison to experimental data Adopted from A. Hastings (1997) Population Biology. Concepts and Models. 16
17 3.3 Qualitative analysis The goal of qualitative analysis of a DE is to try to understand the dynamical behaviour of the system without having to solve the DE Vector filed of an autonomous DE Def: A DE dx dt = f(x) is called autonomous if the right-handside (rhs) f(x) does not explicitly depend on t. In an autonomous DE, the slope of the solution curve does not change as t varies. Thus, for an autonomous DE, x = dx dt for each given value of x. is uniquely defined Def: Vector field. For an autonomous DE x = f(x), one can calculate the value of x for each fixed value of x. One can use an arrow directed at the calculated direction of x in a graph of x(t) vs t, thus generating a filed of arrows called the direction field. On such a direction field, if one traces out a curve that is tangent to each one of the arrows it passes, one would obtain the sketch of a solution curve x(t) in the plot. 17
18 E.g. Sketch the vector field of the DE dn dτ = n(1 n) = f(n). Ans: n = 0.5, dn dτ = 3 4. n = 0.25, dn dτ = n = 0, dn dτ = 0. n = 0.25, dn dτ = n = 0.5, dn dτ = 1 4. n = 0.75, dn dτ = n = 1, dn dτ = 0. n = 1.25, dn dτ = n( τ ) τ n = 1.5, dn dτ =
19 3.3.2 Flows in phase space Def: The space span by model variable(s) is called the phase space (also called the state space). For 1D systems, the phase space is 1D. For 2D systems, the phase space is 2D. In phase space, the independent variable t is not explicitly shown. However, the direction of variations in the state variable can be represented by the flow direction represented by arrows in the phase space. 19
20 E.g. For the logistic equation N = rn(1 N/K) with IC N(0) = N 0, the phase space is the 1D N axis. We know the solution is N(t) = KN 0 N 0 + (K N 0 )e rt. Typical solutions curves are plotted as follows and projected onto the phase space N(0) N(t) N(t) K N(0) K/2 Projecting onto N axis K K/2 N(0) 0 0 t 0 Def: The trace of time variation of the phase variable in phase space is called a solution trajectory. In 1D system, solution trajectories can only be straight arrows pointing either to the left or to the right. 20
21 3.3.3 Steady state of a DE Def: For DE x = f(x), a steady state (also called equilibrium or critical point) is defined as a special value of x = x s such that f(x s ) = 0. AT each steady state x x=xs = f(x s ) = 0, thus the value of x remains unchanged forever (since the rate of change in x is always zero). E.g. For the logistic equation N = rn(1 N/K) = f(n), the steady states are obtained by solving f(n) = rn(1 N/K) = 0 N s = 0, K. Question: Both 0 and K are steady states, but why as t, N(t) K (but not 0) for all N 0 > 0? Answer: Because K is a stable steady state, and 0 is an unstable steady state. Question: How to determine the stability of a steady state? 21
22 3.3.4 Linear stability analysis Consider a 1D DE x = f(x). Let x s be a steady state, i.e. f(x s ) = 0. The stability of x s is determined by the behaviour of the system near x s. Let δ(t) ( δ(t) << 1) be a small difference between x(t) and x s, thus δ(t) = x(t) x s x(t) = x s + δ(t). Substitute into the DE: d(x s + δ) dt = f(x s + δ), δ = f(x s ) + f (x s )δ + O(δ 2 ). Since δ 2 << δ, we can ignore higher order terms in δ and obtain the following linear DE δ = f (x s )δ, δ(t) = δ 0 e f (x s )t. 22
23 Therefore, If f (x s ) > 0, δ(t) as t, x s is unstable. If x s is unstable, phase trajectories (flows) move away from it in at least one directions. If f (x s ) < 0, δ(t) 0 as t, x s is stable. If x s is stable, phase trajectories (flows) converge (move) toward it from all possible directions. If f (x s ) = 0, neutral or stability determined by higher order terms (if exist). Unstable steady states, under normal conditions, cannot be detected in experimental settings or numerical simulations. 23
24 E.g. For the logistic equation N = rn(1 N/K) = f(n), f (N) = r((1 N/K) + rn( 1/K) = r 2rN K. For N s = 0, f (0) = r > 0, thus N s = 0 is unstable. For N s = K, f (K) = r < 0, thus N s = K is stable. E.g. Consider the DE equation x = 3x2 2+x 2 x = f(x) that describes the time evolution of the concentration of a biochemical reactant. (a) Find all steady states. (b) Determine the stability of each steady state found in (a). (c) Determine the long term behaviour of the system for (i) x(0) = 0.5, and (ii) x(0) = 1.5. Answer: (a) (b) f(x) = 3x2 2 + x 2 x = 0 x s1 = 0, and x 2 3x+2 = 0. x s2 = 1, x s3 = 2. f (x) = 3 2x(2 + x2 ) x 2 (2x) (2 + x 2 ) 2 1 = 12x (2 + x 2 )
25 Thus, f (0) = 1 < 0 x s1 = 0 is stable. f (1) = = > 0 x s2 = 1 is unstable. f (2) = = < 0 x s3 = 2 is stable. (c) (i) x(0) = 0.5 is between x s1 = 0 and x s2 = 1, as t, x(t) 0. (ii) x(0) = 1.5 is between x s2 = 1 and x s3 = 2, as t, x(t) 2. 25
26 3.3.5 Steady state(s), stability, phase trajectories, and the plot of f(x) vs x. For a DE of the form x = f(x), the steady states, their stability, and the phase trajectories can all be determined by plotting the graph of f(x) vs x. E.g. For the logistic equation N = rn(1 N/K) = f(n), the graph of f(n) is shown as follows. f(n) f(n) 0 K/2 K N 0 K/2 K N E.g. For the DE x = 3x 2 /(2 + x 2 ) x = f(x), the graph of f(n) is shown as follows. f(x) x
27 E.g. For the DE x = 3x 3 /(2 + x 2 ) x 2 = f(x), the graph of f(n) is shown as follows. f(x) 1 2 x 0 In this case, x s = 0 is neutral based on linear stability analysis. But it is actually unstable (I prefer to call it unstable). Some people would call it half-stable. Def: Phase portrait. A diagram of the phase space in which all steady states are shown together with their stability as well as phase trajectories in each domain of the phase space are clearly demonstrated is called a phase portrait. Def: Basin of attraction. Let x s be a stable steady state of the DE x = f(x). The basin of attraction of x s is defined by an interval B contained in the phase space such that for all x(0) in B, x(t) x s as t. 27
28 (a) A phase portrait obtained for a specific set of parameter values presents a complete picture of the qualitative behaviour of the system for that set of parameters. (b) In a DE with multiple stable steady states, their basins of attraction are typically separated by an unstable steady state. E.g. For x = 3x2 2+x 2 x = f(x), the steady states x s1 = 0 and x s3 = 2 are stable and x s2 = 1 is unstable. The Basin of attraction for x s1 = 0 is (, 1) and the basin of attraction for x s3 = 2 is (1, ). Obviously, the unstable steady state x s2 = 1 separates the two basins of attraction. 28
29 Graphic method: For 1D dynamical systems of the form x = f(x), if one can sketch the graph of the RHS function f(x) vs x, then (a) Each zero of the function (i.e. where the curve of f(x) intersects the horizontal axis) represents a steady state. (b) The linear stability of each steady state is determined by the slope of f(x) at that point. (c) The phase trajectory in each interval between neighbouring steady states is determined by the sign of f(x) in that interval: arrow pointing to the right if f(x) > 0 (i.e. the curve is above the horizontal axis); arrow pointing to the left if f(x) < 00 (i.e. the curve is below the horizontal axis). For logistic model: N = rn(1 N K ) = f(n) f(n) 0 K/2 K N 29
30 For: x = 3x 2 /(2 + x 2 ) x = f(x) f(x) x For: x = 3x 3 /(2 + x 2 ) x 2 = f(x) f(x) x
31 3.4 Further improvement of the logistic model The logistic model N follows = rn(1 N K ) can be expressed as N N = r(1 N K ) = R(N) where the per capita population growth rate or fitness N N = R(N) is a linear decreasing function of N as demonstrated below r N N =r(1 N/K) No Allee effect 0 K N Although the logistic model correctly describes the population dynamics of some experimental systems, the fact that it assumes that the per capita growth rate is highest when the population size the near zero is inconsistent with the observations of many other populations. In these populations, a minimal size of the population is required for the survival and growth of the population. 31
32 Allee effect: In 1930s, biologist Warder Clyde Allee observed that gold fish population grows more rapidly when the population size increases from a low levels. Allee effect refers to the influence of the population size on the effective or per capita growth rate, N N, such that maximum growth occurs at lower to intermediate levels of N but not at N = 0. Per capita growth rate N N Weak No effect Strong Allee effect Allee threshold Population size or density N Allee, WC, Emerson, AE, Park, O, Park, T and Schmidt, KP (1949). Principles of animal ecology. Odum, E, Brewer, R and Barrett, GW (2004). Fundamentals of Ecology. 32
33 A model with Allee effect (modified from Strogatz 2.3.4) N N = r a(n b)2, (r R, a, b > 0 are constants). Or N = N[r a(n b) 2 ]. Questions: 1. Under what conditions does the system show strong Allee effect? 2. Calculate the Allee threshold in terms of parameters r, a, b. 3. Under conditions obtained in (1), find all fixed points and determined their stability. 4. Sketch the solutions N(t) for different initial conditions. 5. Compare the solutions to those found in logistic equation. 33
34 Conditions for Allee effect Plotting N N against N, we obtain N N =r a(n b)^2 b + r/a r > ab^2 0< r < ab^2 0 r < 0 N b r/a = r a(n b)2 is a downward hyperbola with two zeros b ± r/a. N N Strong Allee effect is achieved when 0 < r < ab 2. Allee threshold is N = b r/a. 34
35 Steady states and their stability Given the RHS of N = N[r a(n b) 2 ] = f(n), steady states are obtained by solving f(n) = 0 = f(n) = N[r a(n b) 2 ] N s = 0, b± r/a. To determine the stability, we calculate f (N) = r a(n b) 2 2aN(N b). Thus, For N s = 0, f (0) = r ab 2 < 0 (since r < ab 2 for strong Allee effect). It is stable. For N s = b r/a, f (b r/a) = 2 ran s > 0. It is unstable. For N s = b + r/a, f (b + r/a) = 2 ran s < 0. It is stable. 35
36 Alternatively, one can plot f(n) against N to yield f(n)=n[r a(n b)^2] 0< r < ab^2 (2b b^2+3/a )/3 N s + 0 N s (2b + b^2+3/a )/3 N Based on the above plot, the solution curves can be qualitatively sketched as follows. N(t) + N s N s 0 0 t 36
37 Summary: Through solving the steady state solutions and determining their stability, we understand the long term behavior of a 1D nonlinear differential equation. By plotting the phase portrait, we determine the direction of flows at each point of the phase space. Stable steady states are points of convergence of the phase flows; an unstable steady state often divides the domains of attraction (called the basins of attraction) for different stable fixed points. Combining the results mentioned above and a sketch of the graph of f(n), we can qualitatively sketch the solutions curves with inflection points accurately located and concavity correctly characterized. 37
38 3.5 Bifurcations in nonlinear systems Definition: x = f(x, r), where x R is the phase variable, r R is a control parameter, f(x, r) is a smooth real-valued function of x and r. For each value of r, there exists a corresponding phase portrait. If for some critical value r = r c, the phase portrait changes qualitatively for r < r c and r > r c (e.g. number or stability of steady states changes), then r = r c is called a bifurcation point. A bifurcation point is often where new solution(s) (sometimes related to more organized pattern(s)) emerges. 38
39 3.5.1 Saddle-node bifurcation Also referred to as fold, turning point, blue-sky,.... E.g. A population with density-independent growth rate and competition is described by where x = r x 2 = f(x, r), x 0 is the population size/density. To study its bifurcation structure, we consider x R, treating it just as one typical mathematical equation. r R is a real-valued parameter. The phase portraits for three different values of r. f(x,r) f(x,r) f(x,r) x x x (a) r<0 (no fixed point) (b) r=0 (one fixed point: 0) (c) r>0 (two fixed points: _+ r ) It is obvious the portrait changes at r = 0, from zero to two fixed points. Thus, r c = 0 is a bifurcation point, called a saddle-node (SN) bifurcation in this case. 39
40 The bifurcation diagram Definition: A diagram that shows how the solutions of a dynamic system change as one (or sometimes more) parameter changes. The bifurcation diagram for x = r x 2 is given below on the left. Often, we stack the representative phase portraits on the bifurcation diagram as shown below on the right. x s r x s 0 r 0 r r Phase portraits stacked 40
41 Criteria for SN bifurcation Thereom: x = f(x, r) has a SN bifurcation point at (x s, r c ) if all the following conditions are satisfied. (SN1) f(x s, r c ) = 0 ( x s is a fixed point at r c = 0), (SN2) f x (x s, r c ) = 0, (SN3) f r (x s, r c ) 0, (SN4) 2 f x 2 (x s, r c ) 0. E.g Show that ẋ = r x e x = f(x, r) has a SN bifurcation point at (x s, r c ) = (0, 1). Ans: (SN1) f(x s, r c ) = 1 0 e 0 = 0. Yes. (SN2) f x = 1 + e x (0,1) = 1 + e 0 = 0. Yes. (SN3) f r = 1 0. Yes. (SN4) 2 f x 2 = e x (0,1) = e 0 = 1 0. Yes. Questions: How does the bifurcation diagram look like? (Exercise for you at home?) 41
42 3.5.2 Transcritical bifurcation E.g. Consider the logistic equation with K = 1 x = rx x 2 = x(r x) = f(x, r), for x, r R. The phase portraits for three different values of r. f(x,r) f(x,r) f(x,r) x x x (a) r<0 (b) r=0 (c) r>0 It is obvious the portrait changes at r = 0. Thus, r c = 0 is a bifurcation point, called a transcritical (TC) bifurcation in this case. 42
43 The bifurcation diagram for ẋ = rx x 2 is x s x =r s x s x =0 s 0 r 0 r Criteria for TC bifurcation Thereom: x = f(x, r) has a TC bifurcation point at (x s, r c ) if all the following conditions are satisfied. (TC1) f(0, r) = 0 ( x s = 0 is a fixed point at r c = 0) for all r, (TC2) f x (0, r c) = f x (0, r c ) = 0, (TC3) 2 f x r (0, r c) = f xr (0, r c ) 0, (TC4) 2 f x 2 (0, r c ) = f xx (0, r c ) 0. 43
44 E.g Show that x = r ln x + x 1 has no fixed point at x s = 0 but x s = 1 is a fixed point for all r. Show that there exists a TC bifurcation for this equation and find the critical value r c. Introduce a new variable y = x 1 (i.e. x = y + 1) and substitute it into the equation, we obtain y = r ln(y + 1) + y = f(y, r). Let us check the conditions for TC. (TC1) f(0, r) = = 0 is a fixed point for all r. Yes. (TC2) f y = r c = 1. r y (0,r) = r + 1 = 0 if r = 1. Yes and (TC3) f yr = 1 y+1 (0,r c ) = 1 0. Yes. r (TC4) f yy = (y+1) 2 (0, 1) = 1 0. Yes. Question: How does the bifurcation diagram look like? (Exercise for you at home?) 44
45 3.5.3 Pitchfork bifurcation E.g. Consider the population model in with competition at higher density is described by a cubic term. for x, r R. x = rx x 3 = x(r x 2 ) = f(x, r), Note that the equation is invariant when replacing x by x (symmetry)! This implies that if x s is a fixed point, x s must also be a fixed point. Similarly, if x(t) is a solution x(t) must also be solution. The phase portraits for three different values of r. f(x,r) f(x,r) f(x,r) x x x (a) r<0 (b) r=0 (c) r>0 The portrait changes at r c = 0. It is called a pitchfork (PF) bifurcation 45
46 The pitchfork bifurcation diagram x s r 0 0 r r Criteria for PF bifurcation Thereom: x = f(x, r) has a PF bifurcation point at (0, r c ) if all the following conditions are satisfied. (PF1) f( x, r) = f(x, r) for all x, r. (PF2) f x (0, r c ) = 0. (PF3) f xr (0, r c ) 0, (PF4) f xxx (0, r c ) 0. 46
47 E.g A model for magnets and neural networks. x = x + r tanh x. Show that there exists a PF bifurcation for this equation as r is varied. Find the critical value r c. Ans. Let us check the conditions for PF. (PF1) f( x, r) = x + r tanh( x) = (x + r tanh x) = f(x, r) for all x, r. Yes. (PF2) f x (x, r) = 1 + rsech 2 x (0, r) = 1 + r = 0 if r = 1. Yes and r c = 1. (PF3) f xr (x, r) = sech 2 x (0,1) = 1 0. Yes. (PF4) f xxx (x, r) = r 6 4 cosh2 x cosh 4 x (0,1) = 2 0. Yes. Question: How does the bifurcation diagram look like? (Exercise for you at home?) 47
48 Taylor expansion and bifurcation type Let (x s, r c ) be a bifurcation point of the nonlinear system x = f(x, r). Using the Taylor expansion for multi-variable functions f(x, r) = f(x s, r c ) + f x (x s, r c )(x x s ) + f r (x s, r c )(r r c )+ 1 2 [f xx(x s, r c )(x x s ) 2 + 2f xr (x s, r c )(x x s )(r r c )+ f rr (x s, r c )(r r c ) 2 ] +, E.g Revisited! Let (x s, r c ) be the bifurcation point of x = r ln x + x 1 = f(x, r). For this equation f x (x s, r c ) = r c x s + 1, f xr (x s, r c ) = 1 x s, f xx (x s, r c ) = r c f r (x s, r c ) = ln x s, f rr (x s, r c ) = 0. Thus, taking into account of the fact f(x s, r c ) = 0 f(x, r) [ r c + 1](x x s ) + ln x s (r r c )+ x s 1 2 [ r c (x x x 2 s ) (x x s )(r r c )]. s x s 48 x 2 s ;
49 Also, the steady state that exits irrespective of the parameter r for this equation is x s = 1. Substitute into the equation f(x, r) [r c + 1](x 1) [ r c(x 1) 2 + 2(r r c )(x 1)] = (r + 1)(x 1) r c 2 (x 1)2. Therefore, the approximate DE of the original DE near the bifurcation point is given by x (r + 1)(x 1) r c 2 (x 1)2. Introducing a new state variable y = x 1, a new time variable τ = r c t/2, and a new parameter p = 2(r + 1)/r c, we obtain dy dτ = py y2 or y = py y 2. This is the identical to the standard form (often called the normal form) of the TC bifurcation x = rx x 2. For this DE, the TC bifurcation happens at p c = 0, thus p c = 2(r c + 1)/r c = 0 r c = 1 for the original model. 49
50 Application to the model with Allee effect E.g x = x[r a(x b) 2 ] = x[r (x 1) 2 ] = f(x, r), x, r R. For x s = 0, f (0) = r 1 stable if r < 1, unstable if r > 1. For x s = 1 + r (r > 0), f (1 + r) = 2 r(1 + r) stable for all r > 0. For x s = 1 r (r > 0), f (1 r) = 2 r(1 r) stable for all 0 < r < 1, unstable for r > 1. The bifurcation diagram with all fixed points x s 1 + r r r 50
51 Summary: (1) Stereotypical forms (often referred to as normal forms) of three types of bifurcation points frequently encountered are studied. (2) Criteria for each bifurcation are given together with the corresponding bifurcation diagram. (3) The underlying theorems can be understood in terms of Taylor series expansion of the RHS of the differential equation. (4) Once the bifurcation diagram is obtained, phase portraits of the system for all possible values of the parameter can be sketched based on the diagram. (5) Softwares (such as AUTO or XPPAUT) exist that help calculate some basic bifurcation diagrams of nonlinear differential equations. 51
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