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In this extension, we consider the nation's armature expenditure with the removal of surveillance expenditures of each nation. The purpose behind this being surveillance can be complex like aerial surveillance or simpler such as data mining and profiling which is not an armature expense. Hence we remove all surveillance from the study of armature expenditure. We consider a logistic version of the Richardson model as follows,

$K_p$ = Maximum expenditure of Purple nation on armature

$K_g$ = Maximum expenditure of Green nation on armature

$v_p$ = Surveillance expenditure outsourced, Purple nation

$v_g$ = Surveillance expenditure outsourced, Green nation

$A_p$ = $K_p - v_p$

$A_g$ = $K_g - v_g$

We begin with the "logistic" version of the Richardson model.

$\frac{dx}{dt} = a(1-\frac{x}{A_p})y - mx + r$

$\frac{dy}{dt} = a(1-\frac{y}{A_g})x - ny + r$

where $a,b,m,n > 0$

Solving for the equilibrium,

$ay - \frac{xy^{*}}{A_p} -mx^{*} + r = 0$

$bx - \frac{xy^{*}}{A_g} -ny^{*} + s = 0$

We end up with the following equilibrium values,

$(x_1,y_1) = (\frac{r}{m},0)$

$(x_2,y_2) = (0,\frac{s}{n})$

Determining the Jacobian matrix we attain,

$J(x,y) = \begin{pmatrix}
\frac{-y}{A_p} - m & a - \frac{x}{A_p} \\
b - \frac{y}{A_g} & \frac{-r}{mA_g} - n 
\end{pmatrix}$

We can now perform phase plane analysis where $r,s > 0$. Positivity of these parameters means that there are mutual grievances, that is each nation is increasing armature expenditure.

Case 1:

$J(\frac{r}{m},0) = \begin{pmatrix}
-m & a - \frac{r}{mA_p} \\
b & \frac{-r}{mA_g} - n 
\end{pmatrix}$

$tr(J) = -(m + \frac{r}{mA_g} + n)$

$det(J) = (mn-ab) + r(\frac{1}{A_g} + \frac{b}{mA_p})$

Given all our parameters are positive values $tr(J) < 0$. There are a few situations when it comes to the $det(J)$.

i.   $mn-ab > 0$ implies $det(J) > 0$, stable

ii.  $mn-ab < 0$ where $|{mn-ab}| < r(\frac{1}{A_g} + \frac{b}{mA_p})$ implies $det(J) > 0$, stable

iii. $mn-ab < 0$ where $|{mn-ab}| > r(\frac{1}{A_g} + \frac{b}{mA_p})$ implies $det(J) < 0$, unstable

Thus when scenarios i and ii hold, we have negative real eigenvalues and stability for equilibrium values $(\frac{r}{m},0)$. In scenario iii our equilibrium is unstable.

Case 2:

$J(0,\frac{s}{n}) = \begin{pmatrix}
\frac{-s}{nA_p} - m & a \\
b - \frac{s}{nA_g} & -n 
\end{pmatrix}$

$tr(J) = -(n + \frac{s}{nA_p} + m)$

$det(J) = (mn-ab) + s(\frac{1}{A_p} + \frac{a}{nA_g})$

Given all our parameters are positive values $tr(J) < 0$. There are a few situations when it comes to the $det(J)$.

i.   $mn-ab > 0$ implies $det(J) > 0$, stable

ii.  $mn-ab < 0$ where $|{mn-ab}| < s(\frac{1}{A_p} + \frac{a}{nA_g})$ implies $det(J) > 0$, stable

iii. $mn-ab < 0$ where $|{mn-ab}| > s(\frac{1}{A_p} + \frac{a}{nA_g})$ implies $det(J) < 0$, unstable

Thus when scenarios i and ii hold, we have negative real eigenvalues and stability for equilibrium values $(0,\frac{s}{n})$. In scenario iii our equilibrium is unstable.

In a similar analysis we now consider $r,s < 0$. Negativity of these parameters suggests that there is the good will effect, that is each nation is decreasing armature expenditure.

Case 1:

Given all our parameters are positive values $tr(J) < 0$. There are a few situations when it comes to the $det(J)$.

i.   $mn-ab < 0$ implies $det(J) < 0$, unstable

ii.  $mn-ab > 0$ where ${mn-ab} < |r(\frac{1}{A_g} + \frac{b}{mA_p}|)$ implies $det(J) < 0$, unstable

iii. $mn-ab > 0$ where ${mn-ab} > |r(\frac{1}{A_g} + \frac{b}{mA_p}|)$ implies $det(J) > 0$, stable

Thus when scenario iii holds, we have negative real eigenvalues and stability for equilibrium values $(\frac{r}{m},0)$. In scenarios i, ii our equilibrium is unstable.

Case 2:

Given all our parameters are positive values $tr(J) < 0$. There are a few situations when it comes to the $det(J)$.

i.   $mn-ab < 0$ implies $det(J) < 0$, unstable

ii.  $mn-ab > 0$ where ${mn-ab} < |s(\frac{1}{A_p} + \frac{a}{nA_g})|$ implies $det(J) < 0$, unstable

iii. $mn-ab > 0$ where ${mn-ab} > |s(\frac{1}{A_p} + \frac{a}{nA_g})|$ implies $det(J) > 0$, stable

Thus when scenario iii holds, we have negative real eigenvalues and stability for equilibrium values $(0,\frac{s}{n})$. In scenarios i and ii our equilibrium is stable.

Implications:

The model can be divided into the occurrence of Mutual Grievances and the Good Will Effect.

We start by understanding the meaning of mn-ab which appears consistently throughout our results. It is the relationship of the rate of change in expenditure in both nations differenced on the rate of increase/decrease of a nation's armaments in response to the other nations level. We know a, b is directly proportional to their opposing nation, and m,n is negatively proportional to the nation's own expenditure. More simply, mn is the rate of change in the expenditure of both nations and ab is the rate of change of each nation's armaments relative to the other. It is a comparison of the growth of GDP to armaments. If GDP is growing faster than armaments we have a positive relationship, conversely if armaments grow faster than GDP we have a negative relationship. When both grow proportionally the cases become easier to interpret and is completely dependent on the sign of $r,s$ alone.

Mutual Grievances:

Our model predicts when mutual grievances are present and GDP growth is larger than armature expenditure, we can expect stability to develop given the green nation doesn't spend on armature and the purple nation spends $\frac{r}{m}$. Where $\frac{r}{m}$ is positive expenditure.

Goodwill Effect:

Our model predicts when goodwill is present and GDP growth is larger than armature expenditure, we can expect stability to develop given the purple nation doesn't spend on the armature and the green nation spends $\frac{s}{n}$. Where $\frac{s}{n}$ is negative expenditure.

Relativity to Surveillance:

Remember the relationships

$A_p$ = $K_p - v_p$

$A_g$ = $K_g - v_g$

These are each nation's expenditure costs on armature excluding their individual surveillance costs. Using our previous analysis this tells us,

1. The higher surveillance spending, the smaller our $A_p$ or $A_g$ value. Given the situation of mutual grievances, this implies $s(\frac{1}{A_p} + \frac{a}{nA_g})$ and $r(\frac{1}{A_g} + \frac{b}{mA_p})$ will be much larger thus making stability easier to achieve given constant GDP and armature expenditure growth.

2. The higher surveillance spending, the smaller our $A_p$ or $A_g$ value. Given the situation of goodwill thinking, this implies $s(\frac{1}{A_p} + \frac{a}{nA_g})$ and $r(\frac{1}{A_g} + \frac{b}{mA_p})$ will be much smaller thus making stability easier to achieve given constant GDP and armature expenditure growth.

In both cases, higher surveillance spending is a benefit when it comes to stability. We can theorize that the more a nation spends on surveillance given the previously described conditions for stability, the more consistent that nation's relationship will be to the opposing nation. That is there are fewer erratic responses on either nation's side. This supports easier predictability of nations responses to another creating a feedback loop of stable responses given they choose them.

Fri, 02 Apr 2021 20:44 GMT