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Copyright CRC Press 2023 - present CRC Press

Author: Dr Stephen Lynch National Teaching Fellow FIMA SFHEA

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Chapter 6: Biology

Question 6.1: This is a one-dimensional discrete system:

$$c_{n+1}=f(c_n)=(1-a)c_n+bc_n^re^{-sc_n},$$

where $c_n$ is blood cell count.

Fixed points of period one satisfy the equation: $c_{n+1}=c_n=c$, say.

In this case, solve the equation $f(c)-c=0$, to determine the $c^*$ fixed points.

Suppose that there is a fixed point at $c=c^*$. Then, fixed points are stable if:

$$\left| \frac{df(c^*)}{dc} \right|<1.$$

Label the critical points: $O=(0,0)$, $A=(4,0)$ and $B=(\frac{5}{4},\frac{11}{4})$.

The eigenvalues are:

For critical point O: $\lambda_1=4$, $\lambda_2=-1$, and the critical point is unstable, since one eigenvalue is positive.

For critical point A: $\lambda_1=11$, $\lambda_2=-4$, and the critical point is unstable, since one eigenvalue is positive.

For critical point B: $\lambda_1=-2-3.1225i$, $\lambda_2=-2+3.1225i$, and the critical point is stable, as both eigenvalues have negative real part.

Phase portrait of a predator-prey system. The axes would be scaled by large numbers in applications and the species would co-exist with scaled populations of $x=\frac{5}{4}$ and $y=\frac{11}{4}$, assuming of course that neither of the populations are zero.

Chapter 7: Chemistry

Thus, the balanced chemical reaction equation is:

$$3 \mathrm{NaHCO}_3+\mathrm{H}_3\mathrm{C}_6\mathrm{H}_5\mathrm{O}_7 \rightleftharpoons \mathrm{Na}_3\mathrm{C}_6\mathrm{H}_5\mathrm{O}_7+3\mathrm{H}_2\mathrm{O}+3\mathrm{C}\mathrm{O}_2.$$

Chapter 8: Data Science

The optimal solution is, $z=268$, with $\left(x_1,x_2,x_3 \right)=1.8,20.8,1.6$.