2 Discrete growth models, logistic map (Murray, Chapter 2)

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1 2 Discrete growth models, logistic map (Murray, Chapter 2) As argued in Lecture 1 the population of non-overlapping generations can be modelled as a discrete dynamical system. This is an example of an inherently discrete dynamical system. Discrete systems can also be obtained from continuous ones: Strobing Obtain discrete dynamics by strobing continuous flow at a sequence of times. Using surface of section Consider an n-dimensional continuous system and form an n 1-dimensional surface of section Γ. Construct an orbit from all crossings of a trajectory with Γ (Poincaré map from x τ to x τ+1 ). Example for n = 3: Time discretisation of continuous system N = f(n) to for example implement a continuous model on a computer. Get N(t + δt) N(t) + δtf[n(t)]. In terms of τ = t/δt we obtain a discrete dynamical system N τ+1 = N τ + δtf[n τ ] }{{} F (N τ ). (1) 1

2 Note that all continuous dynamical systems can be mapped on discrete dynamical systems, but the opposite is not true: Not all discrete systems corresponds to a discretization of a continuous system. 2.1 Discrete dynamics (M2.1,2.2) Consider the system N τ+1 = F (N τ ) with some map on the form F = N τ + N τ that preserves N τ 0. What are reasonable shapes of the map F for a growth model? For small population sizes we assume linear growth F (r + 1)N. Due to self-regulation when the system becomes overcrowded, we expect F to reach some maximum and then decay for large population sizes: Note that the population increases if N τ+1 = F (N τ ) is larger than N τ. Graphically this translates into being in a region where F (N τ ) lies above the line F (N τ ) = N τ. Similarly the population decreases if F (N τ ) lies below F (N τ ) = N τ. Points where F (N τ ) cross F (N τ ) = N τ are fixed points (steady states). Applying the map on these points do not change the population. In growth models we always have a fixed point at N τ = 0 that can be either stable (attracting) or unstable (repelling). The dynamics of one-dimensional maps are often visualized using cobweb plots (left panel): 2

3 The dynamics starting at two initial conditions (blue green) are shown. 1. From a value N τ, move vertically to F (N τ ). 2. Move horizontally to the line F (N) = N. The new coordinate is (F (N τ ), F (N τ )) = (N τ+1, N τ+1 ), i.e. we have found the position of N τ+1 on the N τ -axis. 3. Repeat step 1 from N τ+1 This procedure constructs the orbit of the map (right panel above). For the case illustrated there are three fixed points: tow stable at N = 0 and N 2 and one unstable at N 1. Depending on the shape of F the dynamics in the vicinity of a fixed point N changes: 3

4 F (N ) < 1 1 < F (N ) < 0 0 < F (N ) < 1 1 < F (N ) We have four distinct cases depending on the eigenvalue Λ F (N ): Unstable Stable Stable Unstable oscillations oscillations no oscillations no oscillations Λ F (N ) C.f. the fixed points of a continuous dynamical system: stable if f (N ) < 0 and unstable if f (N ) > 0, oscillations are not possible in the continuous system because trajectories cannot intersect. The global dynamics of a discrete one-dimensional system can be chaotic if all fixed points are unstable and if the dynamics is bounded: 4

5 2.2 Bifurcations (M2.2,2.4) Ecological growth models exhibit at least one control parameter, for example the growth rate r. As the parameters change, the system may undergo a series of bifurcations where fixed points change stability. Bifurcations occur as the eigenvalue Λ = F (N ) passes through +1 or through 1 upon variation of the control parameters. The following are examples of bifurcations with Λ = +1. Saddle-node bifurcation (called tangent bifurcation in Murray). Supercritical pitchfork bifurcation: Subcritical pitchfork bifurcation: Transcritical bifurcation: Stability changes as two fixed points move through each other : 5

6 Bifurcations with Λ = +1 have counterparts in continuous dynamical systems: the time discretization of a continuous system (1): F (N τ ) = N τ + δtf(n τ ) gives F (Nτ ) = 1 + δtf (Nτ ). Thus, as the stability exponent λ = f (Nτ ) passes zero, the eigenvalue Λ = F (Nτ ) passes +1. In addition to the examples above, discrete systems may also have bifurcations as Λ passes 1. One example is the period-doubling bifurcation: a stable fixed point transforms into an attracting periodic cycle. Such bifurcations are frequent in the logistic map and other discrete growth models. 2.3 Logistic map (M2.3) The logistic map is a discrete system having the same rate of change of population as the logistic equation from Lecture 1 ( N τ+1 = N τ + rn τ 1 N ) τ. K Comparison to logistic equation The following figure shows a numerical comparison of continuous and discrete logistic models for two growth rates (N 0 = 10, K = 750): 6

7 N(t)K r = 0.2 N(t)K r = 2.2 t t The two models agree for small values of r. For larger values of r there can be significant difference. In the example shown the discrete model show oscillations with period 2. A detailed analysis (below) shows that the discrete map does have a fixed point at N = K, but it is unstable when r = 2.2. Instead the dynamics is attracted to a stable period-two cycle. The inherent time delay of discrete models often give rise to unsteady or oscillatory behaviour (c.f. continuous delay models in Lecture 1) Analysis Introduce a dimensionless population size u τ N τ r/(k(r + 1)) and define ρ r + 1 in the logistic map: u τ+1 = ρu τ (1 u τ ). (2) Assume 0 < u 0 < 1 and 0 < ρ 4 (population size may become negative if ρ > 4). The dimensionless logistic map (2) has two fixed points with corresponding eigenvalues Λ: u (1) = 0 u (2) = 1 1 ρ Λ = F (0) = ρ ( Λ = F 1 1 ) ρ = 2 ρ. For 0 < ρ < 1, u (1) = 0 is stable and the first bifurcation (transcritical) happens at ρ = 1 where u (1) = 0 becomes unstable. 7

8 uτ+1 u u τ ρ For 1 < ρ < 3, u (1) = 0 is unstable and u (2) is stable. Λ = 1 for u (2) when ρ = 3 and the system has no stable fixed points when ρ > ρ 0 = 3. In order to understand the bifurcation, we must consider higher-order iterates of the map as follows. In general the points on an orbit are obtained by successive application of the map: u 1 = F (u 0 ) u 2 = F (F (u 0 )) F 2 (u 0 ). u τ = F τ (u 0 ). For the second iterate of F : u τ+2 = F 2 (u τ ) = ρ [ ρu τ (1 u τ ) }{{} u τ+1 ]( ) 1 ρuτ (1 u τ ) }{{} u τ+1 (3) Denote fixed points of the map F 2 by u. When ρ passes through 3, F 2 (u τ ) undergoes a pitchfork bifurcation: uτ+2 u u τ ρ 8

9 Note that all fixed points of F (u τ ) must also be fixed points of F 2 (u τ ), u (1) u (1) = 0 and u (2) u (2) = 1 ρ 1, but the eigenvalue may be different. As we saw above, when ρ passes 3, the eigenvalue Λ F (u (2)) = 2 ρ decreases through 1 and u (2) becomes unstable. At the same time F 2 (u (2)) = F (F (u)) u = F (F (u (2)))F (u u (2) }{{} (2)) = [F (u (2))] 2 = Λ 2 u (2) increases through Λ 2 = ( 1) 2 = +1 and two new stable fixed points u (3),(4) of F 2 (u τ ) are created in a pitchfork bifurcation: u (3),(4) = ρ + 1 ± (ρ 3)(ρ + 1). 2ρ These are stable (if ρ is not too large) and satisfies F 2 (u (3)) = u (3) and F 2 (u (4)) = u (4) but they are not fixed points of F : F (u (3)) u (3) and F (u (4)) u (4). It follows that F (u (3) ) = u (4) and F (u (4) ) = u (3), i.e. the system approaches a stable period-two cycle u (3), u (4), u (3), u (4),... In conclusion: exactly when the period-one fixed point u (2) = 1 ρ 1 becomes unstable a new period-two cycle is created (period-doubling bifurcation), c.f. Section Note that the eigenvalues of u (3),(4) must be equal because they form a cycle for the original map: Λ 3 F 2 (u (3)) = u F (F (u)) u (3) = F (F (u }{{ (3)))F (u } (3)) = F (u u (4) (3))F (u (4)). 9

10 In the same way Λ 4 F 2 (u (4)) = F (u (3))F (u (4)). As ρ passes ρ the eigenvalues Λ 3 = Λ 4 simultaneously pass 1 and another period-doubling bifurcation occurs (four new stable fixed points are formed in the map F 4 ), creating an attracting periodfour cycle in the original map (stable up to ρ ): uτ+4 u u τ ρ Subsequent period doubling bifurcations give rise to a period-doubling cascade, consisting of bifurcation values ρ 0, ρ 1, ρ 2,.... The stable attracting cycle is obtained by plotting large-time iterates of the map: uτ uτ ρ ln(ρ ρ) The figure shows the attractor (long term limiting behaviour for most initial conditions) of the logistic map for different values of ρ. Unstable fixed points and cycles are only visited for a discrete set of isolated initial conditions and are not shown. Feigenbaum (1978) showed that ρ n ρ n 1 ρ n+1 ρ n = δ 10

11 as n. Also ρ n ρ as n. These results imply ρ ρ n δ n. Extending the range up to ρ = 4: uτ ρ reveals complicated dynamics: when ρ > ρ attractors are either periodic cycles or aperiodic attractors. For ρ = 4 the aperiodic attractor fills the entire interval. For other values of ρ the aperiodic attractor splits to fill a number of smaller intervals, e.g. 2 bands at ρ Such aperiodic attractors are characterized by 1. Apparently irregular orbit 2. Re-appearance of the full pattern in small windows (fractal structure): uτ ρ 11

12 3. Enhanced sensitivity to changes in initial condition u 0 This behaviour is called deterministic chaos: the dynamics appears irregular despite being deterministic. More precisely: Deterministic chaos describes aperiodic, bounded deterministic dynamics with enhanced sensitivity to small changes of initial conditions. The period-doubling cascade leading to chaotic behavior shown here is a common route to chaos. It shows several universal features, for example δ = takes the same value for many systems. 2.4 Higher dimensional discrete maps The graphical analysis for the one-dimensional discrete systems of the previous sections is here complemented with an analytical formulation, valid also for higher dimensions. Consider a general map x τ+1 = F (x τ ). What happens to a small perturbation x + δx after many iterations of a map? Let F n (x) F (F ( F (x))) denote a map applied n times. First evaluate the derivative of F n (x) at a fixed point x. First d = 1: x F n (x) = x=x x F (F n 1 (x)) x=x = F x (F n 1 (x )) [ ] }{{} x F n 1 (x) F n = x (x ) General dimension x F n (x) = [J(x )] n, x=x with J the Jacobian F / x. x x=x 12

13 to first order we have F n (x + δx) F n (x ) + }{{} x F n (x ) δx = x + [J(x )] n δx x }{{} [J(x )] n Let Λ i and e i be eigenvalues and eigendirections of J(x ): It follows J(x )e i = Λ i e i. [J(x )] n e i = Λ n i e i. By decomposition of δx in terms of e i, i.e. δx = i a i e i it follows [J(x )] n δx = i a i Λ n i e i. Eigendirections with Λ i < 1 contracting for large n Eigendirections with Λ i > 1 expanding for large n Eigendirections with Λ i = 1 marginal for large n The fixed point is stable if all Λ i < 1. If any Λ i > 1 the fixed point is unstable p-cycles As found above, discrete maps may show periodic motion. The fixed points of a map applied twice: N τ+2 = F (F (N τ )) are 2-cycles i.e. the orbit is N 0, N 1, N 0,.... Any fixed point of the map satisfies F p (N ) = N for any p. Fixed points of N τ+p = F p (N t ) are cycles of period p. The shortest cycle of N τ is called prime cycle (p-cycle) 13

14 Summary: Comparison maps and flows - Flow f Map F System Continuous ODE Discrete difference equations ẋ = f(x) xτ+1 = F (xτ) Solution Trajectory x(t) Orbit {x0, x1,... } Periodic solution Closed orbit x(t) = x(t + T ) Cycle xτ+t = xτ (fixed point of F T (xτ)) Fixed point cond. f(x ) = 0 F (x ) = x Chaos possible If d > 2 d = 1 if non-invertible map otherwise d > 1 Jacobian Jij = fi/ xj Jij = Fi/ xj Area preserving if ẋ = trj = 0 det J = 1 Eigenvalues at fixed point λi (Re[λi] determines stability) Λi ( Λi determines stability) Stable node (2D) λ1 < λ2 < 0 and Im[λ1,2] = 0 Λ1 < Λ2 < 1 and Im[Λ1,2] = 0 Unstable node (2D) λ1 > λ2 > 0 and Im[λ1,2] = 0 Λ1 > Λ2 > 1 and Im[Λ1,2] = 0 Saddle (unstable) (2D) λ1 < 0 < λ2 and Im[λ1,2] = 0 Λ1 < 1 < Λ2 and Im[Λ1,2] = 0 Stable spiral (2D) λ1,2 = µ ± iω and µ < 0 Λ1,2 = ρe ±iθ and ρ < 1 Unstable spiral (2D) λ1,2 = µ ± iω and µ > 0 Λ1,2 = ρe ±iθ and ρ > 1 Center (2D) λ1,2 = ±iω Λ1,2 = e ±iθ 14