Confusion about Lotka_Volterra model for two competing species

Question

Lotka-Volterra model for two competing species in Boyce's Elementary Differential Equations and Boundary Value problems' is given by: \begin{align*} \frac{dx_{1}(t)}{dt}&=r_{1}x_{1}-a_{11}x_{1}x_{1}-a_{12}x_{1}x_{2} \\ &=r_{1}x_{1}(1-\frac{a_{11}}{r_{1}}x_{1})-a_{12}x_{1}x_{2} \\ &=r_{1}x_{1}(1-\frac{x_{1}}{K_{1}})-a_{12}x_{1}x_{2},\quad x_{1}(0)>0 \\ \frac{dx_{2}(t)}{dt}&=r_{2}x_{2}-a_{22}x_{2}x_{2}-a_{21}x_{1}x_{2} \\ &=r_{2}x_{2}(1-\frac{a_{22}}{r_{2}}x_{2})-a_{21}x_{1}x_{2} \\ &=r_{2}x_{2}(1-\frac{x_{2}}{K_{2}})-a_{21}x_{1}x_{2},\quad x_{2}(0)>0 \\ K_{1}= \frac{r_{1}}{a_{11}}\\ K_{2}= \frac{r_{2}}{a_{22}} \end{align*} $ r_{i}, i=1, 2 $:is the intrinsic growth rate for prey species.

$ a_{ii} $ :are intra-species interference coefficient of the two prey species.

$ K_{i}, i=1, 2 $ : the environmental carrying capacity for ith prey species.

$ a_{ij} $: are inter-species interference coefficient of the two prey species.

While in the book by Murray 'Mathematical Biology: I. An Introduction' is given by: \begin{align*} \frac{dx_{1}(t)}{dt}&=r_{1}x_{1}\left(1-\frac{x_{1}}{K_{1}}-\frac{a_{12}}{K_{1}}x_{2}\right),\quad x_{1}(0)>0\\ \frac{dx_{2}(t)}{dt}&=r_{2}x_{2}\left(1-\frac{x_{2}}{K_{2}}-\frac{a_{21}}{K_{2}}x_{1}\right),\quad x_{2}(0)>0\\ \end{align*} Trying to match them I get \begin{align*} \frac{dx_{1}(t)}{dt}&=r_{1}x_{1}\left(1-\frac{x_{1}}{K_{1}}-\frac{a_{12}}{K_{1}}x_{2}\right) \\ &=r_{1}x_{1}- \frac{r_{1}}{K_{1}}x_{1}x_{1}- \frac{r_{1}}{K_{1}}a_{12}x_{1}x_{2} \\ &=r_{1}x_{1}- a_{11}x_{1}x_{1}- a_{11}a_{12}x_{1}x_{2},\quad x_{1}(0)>0\\ \frac{dx_{2}(t)}{dt}&=r_{2}x_{2}\left(1-\frac{x_{2}}{K_{2}}-\frac{a_{21}}{K_{2}}x_{1}\right) \\ &=r_{2}x_{2}- \frac{r_{2}}{K_{2}}x_{2}x_{2}- \frac{r_{2}}{K_{2}}a_{21}x_{2}x_{1} \\ &=r_{2}x_{2}- a_{22}x_{2}x_{2}- a_{22}a_{21}x_{2}x_{1},\quad x_{2}(0)>0\\ K_{1}= \frac{r_{1}}{a_{11}}\\ K_{2}= \frac{r_{2}}{a_{22}} \end{align*} so here we have an extra factor in the inter-species terms. How they are equivalent?

Answer 1

This is probably not a very satisfactory answer. The very next step in Murray's book (3rd Ed) there is a non-dimensionalised version of the model (eq 3.32) which looks like the one given in Boyce's book (10th Ed) except $\epsilon_i$ and $\sigma_i$ (eq 2, Ch 9). If you plug in $\epsilon_i = \sigma_i = 1$ in Boyce's book you get the same fixed point expression (eq 36) as Murray's book.

The idea is: in the RHS of a $\frac{dx_i}{dt}$ term the $x_i x_i$ represents intra-specific interaction and the $x_i x_j$ represents inter-specific interaction between species $i$ and $j$. You can put constants wherever you like to scale these terms with an interpretation. But the qualitative behavior of the system doesn't change and the coefficients persist in algebra, based on how you plugged them.

But to literally interpret the extra factor in the second set of equations: you can think of them like if $a_{ii}$ is the scaling of intra-specific interference, then $a_{ij}$ is a scaling of $a_{ii}$ for inter-species interaction, i.e. the species $j$ interfere with species $i$ with $a_{ij} a_{ii}$.