Appendices

Appendix 1 – EBM non-exponential growth

Survival of all, survival of the first

When the growth is sub-exponential or super-exponential we obtain a different result from the exponential growth case. Choosing a simple dynamical equation with 2 types:

$\dot{x} = b x^a - \phi x$

$\dot{y} = c y^a - \phi y$

As these are frequencies with constant population, $\dot{x}=-\dot{y}$ and $x+y=1$ , this gives:

$\phi=bx^a + cy^a$

We reduce to one equation in $x$ by substituting in $\phi$ and $y=1-x$ .

$\dot{x}=bx^a - x ( bx^a + c( 1-x )^a )$

$\dot{x}=bx^a ( 1-x ) - cx ( 1-x )^a$

$\dot{x}=bx^{a-1} ( 1-x) x - cx( 1-x )( 1-x )^{a-1}$

$\dot{x}=x( 1-x ) ( bx^{a-1} - c( 1-x )^{a-1} )$

Setting $f(x)=bx^{a-1}-c(1-x)^{a-1}$ we obtain:

$\dot{x}=x(1-x)f(x)$

There are three fixed points of the dynamics, $x^*=0, x^*=1, f(x^*)=0$ . Solving for $f(x^*)=0$

$bx^{*a-1}=c(1-x^*)^{a-1}$

$\sqrt[a-1]{b}x^*= \sqrt[a-1]{c}(1-x^*)$

The three fixed points $x^*$ are

$x^* = 0,\frac{1}{1+\sqrt[a-1]{b/c}},1$

Calculating fixed point stability

Fixed point $x^*=0$

We choose the first f.p. $x^*=0$ . Linearising around that f.p., $\epsilon=x, \dot{\epsilon}=\dot{x}$ :

$\dot{\epsilon} \approx \epsilon (1-\epsilon) f(\epsilon)$

$f(\epsilon) = b\epsilon^{a-1} - c (1-\epsilon)^{a-1}$

$f(\epsilon) \approx b\epsilon^{a-1} -c+c(a-1)\epsilon$

$\dot{\epsilon} \approx (1-\epsilon) ( b\epsilon^a -c\epsilon)$

$\dot{\epsilon} \approx b\epsilon^a -c\epsilon$

If $a<1$ then $\dot{\epsilon}\approx b\epsilon^a$ and the f.p. is unstable. If $a>1$ then $\dot{\epsilon}\approx-c\epsilon$ and the f.p. is stable.

Fixed point $x^*=1$

Linearising around $x^*=1$ , $\epsilon=1-x, \dot{\epsilon}=-\dot{x}$ :

$\dot{\epsilon} \approx -(1-\epsilon) \epsilon f(\epsilon)$

$f(\epsilon) = b(1-\epsilon)^{a-1} - c\epsilon^{a-1}$

$f(\epsilon) \approx b -b(a-1)\epsilon - c\epsilon^{a-1}$

$\dot{\epsilon} \approx -(1-\epsilon) (b\epsilon - c\epsilon^a)$

$\dot{\epsilon} \approx c\epsilon^a -b\epsilon$

We have the same stability analysis as before. If $a<1$ then the f.p. is unstable. If $a>1$ then the f.p. is stable.

Middle fixed point $0<x^*<1$

As this is a 1-dimensional system the middle f.p. must be stable when $a<1$ and unstable when $a>1$ .

Conclusion

When the growth is sub-exponential, or $a<1$ , the middle f.p. is the only stable configuration so there is ‘Survival of all’. Conversely, when the growth is super-exponential or $a>1$ , the middle f.p. is unstable and only $x=0$ or $y=0$ is stable. This is ‘Survival of the first’.

Appendix 2 – Varying population size in non-exponential growth EBM

We assume the form of the linked differential equations to be:

$\dot{X}_i=f_i X^a_i -\psi X_i$

As before, $\dot{X}_i=\dot{P}x_i+P\dot{x}_i$ :

$\dot{P}x_i+P\dot{x}_i=f_i (P x_i)^a-\psi P x_i$

$\dot{x}_i=f_i P^{a-1}x^a_i-\psi x_i-\dot{P}P^{-1}x_i$

Summing across $i$ gives:

$\psi=P^{a-1} \sum f_i x^a_i -\dot{P}P^{-1}$

Then the $\dot{x}_i$ are:

$\dot{x}_i=P^{a-1}f_i x^a_i -P^{a-1} x_i \sum f_i x^a_i$

If we now consider just two types so that we can reduce the differential equations to just one using $x_1+x_2=1$ :

$\dot{x}=P^{a-1}b x^a - P^{a-1} \left( b x^a + c (1-x)^a \right) x$

$\dot{x}=P^{a-1} \left( b x^a(1-x) -c x (1-x)^a \right)$

Setting $f(x)=bx^{a-1} - c(1-x)^{a-1}$ we obtain:

$\dot{x}=x(1-x)f(x)P^{a-1}$

We assume in the first place that $P$ is always strictly positive. Comparing to the constant population equation for $a\neq 1$ , we see that there are the same three fixed points in the same location. We can also see that due to $f(x)$ being the same equation the stability analysis is the same for varying population size to constant population size.

Can ‘Survival of all’ and ‘Survival of the first’ lead to extinction?

We now remove that restriction that $P$ is strictly positive and let $P(x)$ reach zero at one point in the domain $0<x<1$ . Then there are four fixed points of the dynamics, $x^*=0, f(x^*_f)=0, P(x^*_p)=0, x^*=1$ .

We assume that $x_1=x$ is the fittest type associated with extinction and $x_2=1-x$ is the least-fit type associated with extant. Then we have $P=-d x+e y$ . We can safely ignore $x^*=1$ as it is unreachable. That gives three fixed points that we need stability information for.

Before we can analyse their stability we give the solution to the two remaining fixed points.

$P=-d x + e (1-x) = e-(d+e)x$

Solving for $x$ :

$x^*_p=\frac{e}{d+e}$

And from Appendix 1:

$x^*_f=\frac{1}{1+\sqrt[a-1]{b/c}}$

Stability of the fixed point $x^*=0$

Firstly, $x=\epsilon$ and $\dot{x}=\dot{\epsilon}$ where $\epsilon<<1$ :

$\dot{\epsilon}=\epsilon(1-\epsilon)f(\epsilon)P(\epsilon)^{a-1}$

$f(\epsilon)\approx b\epsilon^{a-1} - c + c(a-1)\epsilon$

$P(\epsilon)^{a-1}=\left[e-(d+e)\epsilon\right]^{a-1}$

$P(\epsilon)^{a-1}=e^{a-1}\left[1-\frac{e+d}{e}\epsilon\right]^{a-1}$

$P(\epsilon)^{a-1}\approx e^{a-1}\left[1-(a-1)\frac{e+d}{e}\epsilon\right]$

$\dot{\epsilon}\approx (1-\epsilon)(b\epsilon^a-c\epsilon)\left[1-(a-1)\frac{e+d}{e}\epsilon\right]e^{a-1}$

$\dot{\epsilon}\approx e^{a-1}\left( b\epsilon^a-c\epsilon \right)$

$e^{a-1}$ is a positive constant always. For sub-exponential growth, $a<1$ and $x^*=0$ is unstable. For super-exponential growth, $a>1$ and $x^*=0$ is stable. As this is a 1-dimension differential equation, we know that consecutive fixed points must be of opposite stability.

Conclusion

We illustrate the dynamics of the $(x,y)$ system on the line $0\leq x < 1$ for the three possible $a$ . The derivation of the two non-exponential cases is in Appendix 2.

We saw from $a=1$ that in some circumstances ‘Survival of the fittest’ can change into ‘Survival of none’. This is illustrated in the top figure above.

The question “Can $a<1$ or $a>1$ lead to extinction?” can be answered with “possibly”, if the circumstances are ‘unfortunate enough’. We also see that as one goes from $a>1$ to $a<1$ to $a=1$ , the likely hood of extinction increases. Of course this applies for 2 types, a fuller investigation of many types may find a different conclusion.

Appendix 1 – EBM non-exponential growth

Survival of all, survival of the first

Calculating fixed point stability

Fixed point $x^*=0$

Fixed point $x^*=1$

Middle fixed point $0<x^*<1$

Conclusion

Appendix 2 – Varying population size in non-exponential growth EBM

Can ‘Survival of all’ and ‘Survival of the first’ lead to extinction?

Stability of the fixed point $x^*=0$

Conclusion

Appendix 3 – Explanation of and comparison of agent based and equation based models

Appendix 4 – Assumptions of model 1

Appendix 5 – Assumptions of model 2