Bifurcation Theory

Abstract

Consider two populations, x and y, that are interacting either by competition, or as predator and prey. They may end up near a stable steady state, or possibly in seasonally oscillating states; this could depend on their proliferation rates, death rates, available resources, climate change, etc. In this chapter we wish to explore these varied possibilities using mathematics. To do that we begin by a short introduction to the theory of bifurcations. Bifurcation theory is concerned with the question of how the behavior of a system which depends on a parameter p changes with the parameter. It focuses on any critical value, p = p _cr , where the behavior of the system undergoes radical change; such values are called bifurcation points.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Consider two populations, x and y, that are interacting either by competition, or as predator and prey. They may end up near a stable steady state, or possibly in seasonally oscillating states; this could depend on their proliferation rates, death rates, available resources, climate change, etc. In this chapter we wish to explore these varied possibilities using mathematics. To do that we begin by a short introduction to the theory of bifurcations. Bifurcation theory is concerned with the question of how the behavior of a system which depends on a parameter p changes with the parameter. It focuses on any critical value, p = p _cr , where the behavior of the system undergoes radical change; such values are called bifurcation points. The change that occurs at p = p _c typically involves two or more branches of solutions which depend on the parameter p; the nature of these ‘bifurcation’ branches changes radically at p = p _c .

We shall consider bifurcation phenomena for a system of differential equations with parameter p,

$$\displaystyle{ \frac{d\mathbf{x}} {dt} = \mathbf{f}(\mathbf{x},p). }$$

(11.1)

Bifurcation points can arise in different ways. For example, suppose a steady state of Eq. (11.1), which depends on p, is stable for p < p _c but loses stability at p _c . Then a qualitative change has occurred in the phase portrait of the system (11.1), and p = p _c is a bifurcation point. It sometimes happens that as p increases from p < p _c to p > p _c the differential system will begin to have periodic solutions, a well-recognized biological phenomena. Thus we would like to determine, mathematically, when such a situation takes place.

Problems 11.1–11.3 are simple but typical examples of bifurcations that frequently occur in biology.

FormalPara Problem 11.1.

Consider the equation

$$\displaystyle{\frac{dx} {dt} = p + x^{2}.}$$

It has two steady states $x = \pm \sqrt{-p}$ if p < 0 and no steady states if p > 0. Prove that $x = -\sqrt{-p}$ is stable and $x = +\sqrt{-p}$ is unstable. The point p = 0 is called a saddle-point bifurcation.

FormalPara Problem 11.2.

Consider the equation

$$\displaystyle{\frac{dx} {dt} = px - x^{2}.}$$

It has steady points x = 0 and x = p. Prove that x = 0 is stable if p < 0 and unstable if p > 0, and x = p is unstable if p < 0 and stable if p > 0. Such a point p = 0, where there is an exchange of stability in the branches of the steady points, is called a transcritical bifurcation.

FormalPara Problem 11.3.

Consider the equation

$$\displaystyle{\frac{dx} {dt} = px - x^{3}.}$$

Show that x = 0 and $x = \pm \sqrt{p}$ (for p > 0) are the steady states of this equation, and determine their stability. The point p = 0 is called a pitchfork bifurcation.

Figure 11.1 illustrates the above three examples.

Fig. 11.1:

(a) Saddle-point bifurcation diagram. (b) Transcritical bifurcation diagram. (c) Pitchfork bifurcation. Solid curves represent stable steady states, while dotted curves are unstable steady states.

FormalPara Example 11.1.

Consider a species x with logistic growth whose death rate is a parameter p,

$$\displaystyle{ \frac{dx} {dt} = rx(1 - \frac{x} {K}) - px. }$$

(11.2)

It has two steady states: x = 0 and $x = K(1 -\frac{p} {r})$, but the latter one is biologically feasible only if x > 0, that is, if p < r. The two branches of steady points intersect at p = r where exchange of stability occurs: x = 0 is stable if p > r and unstable if p < r, whereas $x = K(1 -\frac{p} {r})$ is stable if p < r and unstable if p > r. Thus transcritical bifurcation occurs at p = r; see Fig. 11.2.

Fig. 11.2:

Transcritical bifurcation diagram for Eq. (11.2). Solid lines represent stable steady states, while dotted lines are unstable steady states.

When the density of species x is very small (say 0 < x < 1), mating becomes difficult: The probability of a male from x to meet and mate with a female from x is proportional to x × x. Hence instead of growth rates

$$\displaystyle{\frac{dx} {dt} = rx,}$$

we have growth rate

$$\displaystyle{\frac{dx} {dt} = rx^{2}}$$

or, under constraints represented by a carrying capacity K,

$$\displaystyle{\frac{dx} {dt} = rx^{2}(1 - \frac{x} {K}).}$$

FormalPara Example 11.2.

Consider species x with dynamics

$$\displaystyle{ \frac{dx} {dt} = rx^{2}(1 - \frac{x} {K}) - px. }$$

(11.3)

It has three branches of steady points given by x = 0 and

$$\displaystyle{rx(1 - \frac{x} {K}) - p = 0,\mbox{ or }x = \frac{K} {2} \pm \sqrt{\frac{K^{2 } } {4} -\frac{pK} {r}}.}$$

In this example pitchfork bifurcation occurs at $p = \frac{r} {4}K$, as illustrated in Fig. 11.3.

Fig. 11.3:

Pitchfork bifurcation diagram for Eq. (11.3). Solid lines represent stable steady states, while dotted lines are unstable steady states.

11.1 Hopf Bifurcation

We next consider a different type of bifurcation whereby steady points bifurcate into periodic solutions; this of course must involve a dynamical system with at least two equations.

Consider the following system of two equations, with bifurcation parameter p:

$$\displaystyle\begin{array}{rcl} \frac{dx_{1}} {dt} & =& px_{1} -\mu x_{2} - ax_{1}(x_{1}^{2} + x_{ 2}^{2}),{}\end{array}$$

(11.4)

$$\displaystyle\begin{array}{rcl} \frac{dx_{2}} {dt} & =& \mu x_{1} + px_{2} - ax_{2}(x_{1}^{2} + x_{ 2}^{2}),{}\end{array}$$

(11.5)

where μ, a are positive constants. It is easily seen that the point $x_{1} = x_{2} = 0$ is a steady point, stable if p < 0 and unstable if p > 0. But for p > 0 there also exists a periodic solution,

$$\displaystyle{x_{1}(t) = \sqrt{\frac{p} {a}}\cos \mu t,\quad x_{2}(t) = \sqrt{\frac{p} {a}}\sin \mu t,}$$

which traces the circle $x_{1}^{2} + x_{2}^{2} = \frac{p} {a}$ as t varies.

This type of bifurcation, which gives rise to periodic solutions, is called Hopf bifurcation. Fig. 11.4 illustrates the periodic solutions which arise in the Hopf bifurcation. Note that the Jacobian matrix J at the (0, 0), where the bifurcation occurs, is given by

$$\displaystyle{ J = \left (\begin{array}{*{10}c} p&-\mu \\ \mu & p \end{array} \right ), }$$

and the characteristic equation is

$$\displaystyle{(p-\lambda )^{2} +\mu ^{2} = 0,}$$

so that the eigenvalues are

$$\displaystyle{\lambda = p \pm i\mu.}$$

As p crosses from p < 0 to p > 0, the two eigenvalues, at p = 0, become pure imaginary numbers. It is this behavior of the eigenvalues of the Jacobian matrix that gives rise to the periodic solutions. In fact, the bifurcation behavior in the example of the system (11.4)–(11.5) is a special case of the following theorem.

Fig. 11.4:

Hopf bifurcation for Eq. (11.4)–(11.5): periodic solutions with increasing diameter $\sqrt{\frac{p} {a}}$.

Theorem 11.1.

( Hopf Bifurcation ) Consider the system

$$\displaystyle{ \frac{dx} {dt} = f(x,y,p),\quad \frac{dy} {dt} = g(x,y,p). }$$

(11.6)

Assume that for all p in some interval there exists a steady state (x ^s (p),y ^s (p)), and that the two eigenvalues of the Jacobian matrix (evaluated at the steady state) are complex numbers $\lambda _{1}(p) =\alpha (p) + i\beta (p)$ and $\lambda _{2}(p) =\alpha (p) - i\beta (p)$ . Assume also that

$$\displaystyle{\alpha (p_{0}) = 0,\quad \beta (p_{0})\neq 0\quad \mbox{ and }\ \frac{d\alpha } {dp}(p_{0})\neq 0.}$$

Then one of the three cases must occur:

• 1. there is an interval p ₀ < p < c ₁ such that for any p in this interval there exists a unique periodic orbit containing (x ^s (p ₀ ),y ^s (p ₀ )) in its interior and having a diameter proportional to $\vert p - p_{0}\vert ^{1/2}$ ;

• 2. there is an interval c ₂ < p < p ₀ such that for any p in this interval there exists a unique periodic orbit as in case (1);

• 3. for p = p ₀ there exist infinitely many orbits surrounding (x ^s (p ₀ ),y ^s (p ₀ )) with diameters decreasing to zero.

A proof of Theorem 11.1 can be found, for instance, in Reference [6].

For the special system (11.4)–(11.5), p ₀ = 0, (x ^s(p ₀), y ^s(p ₀)) = (0, 0), α(p) = p, β(p) = μ, and case 1 occurs with p > 0 if a > 0 as shown above, case 2 occurs with p < 0 if a < 0, and case 3 occurs if a = 0 as illustrated in Fig. 3.1(F).

Example 11.3.

We consider a model of herbivore–plant interaction. The plant P has logistic growth with carrying capacity K, and the herbivore H has eating capacity $\sigma$, which we take as the bifurcation parameter. Then

$$\displaystyle\begin{array}{rcl} \frac{dP} {dt} & =& rP(1 - \frac{P} {K}) -\sigma \frac{P} {1 + P}H, {}\\ \frac{dH} {dt} & =& \gamma \sigma \frac{PH} {1 + P} -\mu H, {}\\ \end{array}$$

where γ is the yield constant and μ is the death rate of the herbivore. Rewriting these equations in the form

$$\displaystyle\begin{array}{rcl} \frac{dP} {dt} & =& P[r(1 - \frac{P} {K}) - \frac{\sigma H} {1 + P}], {}\\ \frac{dH} {dt} & =& H[\gamma \sigma \frac{P} {1 + P}-\mu ], {}\\ \end{array}$$

we easily compute the nonzero steady state

$$\displaystyle{P = \frac{\mu } {\gamma \sigma -\mu },\quad H = \frac{r} {\sigma } (1 + P)(1 - \frac{P} {K}) = \frac{r\gamma } {\gamma \sigma -\mu }(1 - \frac{\mu } {K(\gamma \sigma -\mu )}),}$$

and, by the factorization rule, the Jacobian is computed to be

$$\displaystyle{J = \left (\begin{array}{*{10}c} P(-\frac{r} {K} + \frac{\sigma H} {(1+P)^{2}} )& &-\sigma \frac{P} {1+P} \\ \frac{\gamma \sigma H} {(1+P)^{2}} & & 0 \end{array} \right ).}$$

The characteristic equation is then

$$\displaystyle{ \lambda ^{2} - a\lambda + b = 0, }$$

(11.7)

where $b =\det J(P,H)> 0$ and a = trace J(P, H) is given by

$$\displaystyle{a = a(\sigma ) = P(-\frac{r} {K} + \frac{\sigma H} {(1 + P)^{2}}).}$$

We compute

$$\displaystyle{\sigma H = \frac{r\gamma [K(\gamma \sigma -\mu )-\mu ]} {K(\gamma \sigma -\mu )^{2}} \sigma,\quad 1 + P = \frac{\gamma \sigma } {\gamma \sigma -\mu }.}$$

Hence

$$\displaystyle{ \frac{\sigma H} {(1 + P)^{2}} = \frac{r[K(\gamma \sigma -\mu )-\mu ]} {K\gamma \sigma } }$$

and

$$\displaystyle\begin{array}{rcl} a(\sigma )& =& \frac{Pr} {K} \{ - 1 + \frac{1} {\gamma \sigma } [K(\gamma \sigma -\mu )-\mu ]\} {}\\ & =& \frac{Pr} {K\gamma \sigma } [(K - 1)\gamma \sigma - (K + 1)\mu ]. {}\\ \end{array}$$

It follows that $a(\sigma _{0}) = 0$ if

$$\displaystyle{\sigma _{0} = \frac{K + 1} {K - 1} \frac{\mu } {\gamma }}$$

and

$$\displaystyle{\frac{da} {d\sigma } \left \vert _{\sigma =\sigma _{0}} = \frac{Pr} {K\gamma \sigma _{0}}(K - 1)\gamma> 0\quad \mbox{ if }K> 1,\right.}$$

so that $a(\sigma ) <0$ if $\sigma <\sigma _{0}$.

We now observe that the points P, H are both positive if

$$\displaystyle{\sigma> \frac{K + 1} {K} \frac{\mu } {\gamma },}$$

and $\sigma _{0}$ satisfies this inequality since

$$\displaystyle{\frac{K + 1} {K - 1}> \frac{K + 1} {K}.}$$

We conclude that

$$\displaystyle{a(\sigma ) <0\quad \mbox{ if }\ \frac{K + 1} {K} \frac{\mu } {\gamma } <\sigma <\sigma _{0},}$$

$$\displaystyle{a(\sigma _{0}) = 0,\quad \frac{d} {d\sigma }a(\sigma _{0})> 0.}$$

Since the eigenvalues of (11.7) are $\lambda _{1,2} = \frac{a} {2} \pm \sqrt{\frac{a^{2 } } {4} - b}$ and b > 0, we see, using Theorem 11.1, that Hopf bifurcation occurs at $\sigma =\sigma _{0}$. Thus as $\sigma$ increases to $\sigma _{0}$ the stable equilibrium $(P(\sigma ),H(\sigma ))$ becomes unstable and, instead, the dynamics of the herbivore–plant model develops periodic solutions with diameters which increase with $\vert \sigma -\sigma _{0}\vert$. Thus both plant and herbivore will coexist, and their populations will vary ‘seasonally.’

11.2 Neuronal Oscillations

Neuronal oscillations are periodic electrical oscillations along the axon of the neurons, and some simplified models represent them in the form

$$\displaystyle\begin{array}{rcl} \frac{dv} {dt} & =& f(v) - w + I, {}\\ \frac{dw} {dt} & =& \varepsilon (\gamma v - w), {}\\ \end{array}$$

where I is the applied current, arriving from dendrites, which triggers the oscillations. The function f(v) is a cubic polynomial and $\varepsilon$ is a small parameter. The diameter of the periodic oscillations depends on f but is independent of the parameter I. Motivated by this model we consider here the case where f is a quadratic polynomial, and show that this case gives rise to Hopf bifurcation, that is, to periodic oscillations which begin with small diameter as I crosses a bifurcation parameter I ₀, and then increase with I, proportionally to $(I - I_{0})^{1/2}$. For simplicity we take f(v) = v ².

Problem 11.4.

Consider a system

$$\displaystyle\begin{array}{rcl} \frac{dv} {dt} & =& v^{2} - w + I, {}\\ \frac{dw} {dt} & =& 2\gamma v - w, {}\\ \end{array}$$

where $\gamma> \frac{1} {4}$ and 0 < I < γ ². Show that the only steady state $(\bar{v},\bar{w})$ is given by $\bar{v} =\gamma -\sqrt{\gamma ^{2 } - I}$, $\bar{w} = 2\gamma \bar{v}$, that it is stable if $I <\gamma -\frac{1} {4}$, and that Hopf bifurcation occurs at $I =\gamma -\frac{1} {4}$.

11.3 Endangered Species

Consider species with very sparse density v, which is endangered as a result of endemic incurable disease caused by a parasite with density w. Since the population of v is spread over a large territory, mating between a male from v and female from v is proportional to v × v = v ². Hence

$$\displaystyle{\frac{dv} {dt} = rv^{2} -\alpha vw,}$$

where α is the rate by which the parasite w depletes v. On the other hand, the growth of the parasite is proportional to v, so that

$$\displaystyle{\frac{dw} {dt} =\gamma v -\beta w,}$$

where β is the death rate of w. If r β −α γ ≠ 0 then the only steady point is $(\bar{v},\bar{w}) = (0,0)$. In order to save the endangered species v from extinction, new population of the species are introduced into the territory, at density rate I, so that

$$\displaystyle{\frac{dv} {dt} = rv^{2} -\alpha vw + I.}$$

This results in steady points $(\bar{v},\bar{w})$ where $\bar{v}> 0$, $\bar{w}> 0$, and the question arises: are these points $(\bar{v}(I),\bar{w}(I))$ stable for all I?

To address this question we take, for simplicity, $r =\alpha =\beta = 1$, and 1 < γ < 2 and consider I as a bifurcation parameter. Then

$$\displaystyle\begin{array}{rcl} \frac{dv} {dt} & =& v^{2} - wv + I, {}\\ \frac{dw} {dt} & =& \gamma v - w. {}\\ \end{array}$$

The only steady point is $\bar{w} =\gamma \bar{ v}$, $\bar{v} = ( \frac{I} {\gamma -1})^{1/2}$, and the Jacobian matrix about $(\bar{v},\bar{w})$ is

$$\displaystyle{J = \left (\begin{array}{*{10}c} (2-\gamma )\bar{v}& &-\bar{v}\\ \gamma & & -1 \end{array} \right ).}$$

Hence $\det J = 2\,\bar{v}\,(\gamma -1)> 0$ and

$$\displaystyle{\mbox{ trace }J = (2-\gamma )( \frac{I} {\gamma -1})^{1/2} - 1 \equiv A(I),}$$

where A(I) < 0 if I < I ₀, A(I) > 0 if I > I ₀, with

$$\displaystyle{I_{0} = \frac{\gamma -1} {(2-\gamma )^{2}}.}$$

The eigenvalues of J are

$$\displaystyle{\lambda =\sigma \pm i\tau,}$$

where $\sigma = \frac{1} {2}A(I)$, $\tau = [(\frac{1} {2}A(I))^{2} - 2(\gamma -1)\bar{v}]^{1/2}$, and $\frac{d\sigma } {dI}> 0$ at I = I ₀. Hence $(\bar{v},\bar{w})$ is a stable steady point if I < I ₀, and Hopf bifurcation occurs at I = I ₀. We conclude that as I is increased, the population $\bar{v}$, in the steady state, will increase and remain stable as long as I < I ₀; thereafter the steady point will become unstable, and the populations of v and w will oscillate periodically.

Problem 11.5.

Consider the following predator–prey model with sparse prey population, x,

$$\displaystyle\begin{array}{rcl} \frac{dx} {dt} & =& x^{2}(1 - x) - xy, {}\\ \frac{dy} {dt} & =& 4xy - 4\alpha y, {}\\ \end{array}$$

where α > 0. It has an equilibrium point (α, α(1 −α)) for any 0 < α < 1. Prove that the equilibrium point is stable if $\alpha> \frac{1} {2}$ and that Hopf bifurcation occurs at $\alpha = \frac{1} {2}$.

The biological interpretation is that if the predator death rate is smaller than 2 then both predator and prey coexist in steady state, but if the predator death rate exceeds 2 then both predator and prey still coexist and their densities vary periodically.

11.4 Numerical Simulations

To plot the bifurcation diagram, one needs to scan through the parameter space and solve the ODEs for those parameters. If we would like to plot the bifurcation diagram for

$$\displaystyle{\frac{dx} {dt} = f(x,p),}$$

the first step is to plot the nullcline on the x-p plane (f(x, p) = 0), which corresponds to the steady states x _s under different p. Next, on the nullcline, we need to determine which part (branch) is stable and unstable. Let us consider the example

$$\displaystyle{\frac{dx} {dt} = x^{2} + p.}$$

First we plot the curve of $x^{2} + p = 0$ on p-x plane. In MATLAB, define the right-hand side function in a script file:

function y = saddlefun(x,p)

y = p + x.^2;

Note that p and x could be matrices in order to accommodate the matrix of discretized mesh grid on p-x space. To plot the bifurcation diagram, we create another function file called ‘bifurcation.m’ (see Algorithm 11.1). The input of this function is the name of the right-hand-side function of the ODE (e.g., ‘saddlefun’) and the ranges of x and p to plot. That is, to run this code, we should type a command similar to the following in the command window:

>> bifurcation(‘saddlefun’,[-5,5],[-5,5])

In ‘bifrurcation.m’, we first discretize x-p plane with a 101 × 101 mesh grid (using the command ‘meshgrid’). Then we try to plot the zeros of x ² + p by using the ‘contour’ command, as shown in Fig. 11.5(A). Next, for each p, we need to start with an initial condition x ₀ which is not a steady state and run until we arrive near a steady state. Therefore, we avoid the x ₀ too close to the nullclines, and use the rest of the points as initial conditions to do time evolution (green points in Fig. 11.5(B)). The solution will move away from the unstable branch and be attracted to the stable branch (blue circles in Fig. 11.5(C)). When we run ‘bifurcation.m,’ we will see Fig. 11.5(A)–(C) consecutively.

Fig. 11.5:

Bifurcation diagram for saddle point bifurcation. (A) The red curve is the nullcline; (B) The green area are covered by green circles, which are non-steady-state points; (C) Solutions marked by blue are all the steady states starting with initial conditions marked by green; that is, only half of the red curve is stable.

Problem 11.6.

Plot the bifurcation diagram for

$$\displaystyle{\frac{dx} {dt} = px - x^{3}.}$$

with range $-5 \leq p \leq 5,-5 \leq x \leq 5$.