Dynamics
What are the dynamics of the system around these non-zero critical pairs of Human and Fish populations? Â Â We begin by setting equations (i) and (ii) equal to zero:
and analyze the local dynamics (local linearization) of the H & F populations around the non-zero critical values Hc and Fc computed in (v) above: for the Human population, we take its partial derivative with respect to Humans and Fish populations around their critical values Hc & Fc; and we then repeat for the Fish population.
This gives is the Jacobian matrix:
To get a sense of the actual dynamics between the two evolving and co-dependent populations, we need to select parameter values for our model. An illustration follows, assuming unconstrained growth rates of 1% and 2% for Human and Fish populations, environmental limitations of 0.8% and 1.9% respectively, and cross-specie encroachment (i.e negative impact from fishing and pollution) of 1% and 1.5% respectively.
With these parameter values, we are now able to specify our model, and calculate the eigenvalues of our system:
Result
 Eigenvalues of opposite sign are characteristics of a saddle path, indicating unstable dynamics between Humans and Fish populations in our model, as specified. Different input parameters result in different (non-saddle path) dynamics, so confidence in the dynamics of our system would require solid empirical support to estimate the input parameters. That said to the extent the parameters used above are realistic, our model would suggest that the environmental dynamics at play today { growing human population -> increasing pollution spilling into primary fishing zones -> combined with rising fishing hauls to feed more humans -> resulting in more toxicity entering the human food chain} are likely to be unstable for both human and fish populations over time, and endogenous corrective forces in the system may not be sufficient to result in a stable long term equilibrium in which both populations survive. At the very least, counting on the current system to self-correct without exogenous intervention would seem reckless, considering the extinction risks of both populations.
Potential solutions
To have staying power, solutions must address the perverse incentive of pollution and over-fishing highlighted in our Prisoner’s Dilemma example:  pollution and over-fishing currently benefit the perpetrators up front, while costs are shared inter-temporally by the entire human and fish populations. Correcting that externality is likely to be an essential element of any stable, long term solution.
Public Solution:   One relatively direct solution would be for the World Trade Organization to appoint an honest, trusted independent third party (Sweden?) with the task of measuring ocean pollution off of the coasts of both Pacific neighbors (e.g. using 10,000 monitoring bouys), reporting the results daily with full transparency and monitoring, and whenever pollution exceeded a certain preset sustainable threshold levels for a period of time (e.g. 10 days?), the offending party would find the price of its export goods taxed in foreign markets, with tax proceeds used by a competent NGO (Cousteau?) to clean up ocean pollution and help revitalize marine life. The economic incentive to pollute would be lowered by the tax on the offending nation’s export profits, and if pollution did not fall below the sustainable threshold, funding would now exist to counter its negative environmental impact. As importantly, there would now be market incentives for businesses on both sides of the Pacific to find ways to grow sales (and contribute to GDP) in more sustainable, environmentally sustainable ways.
Private Solution: Large asset managers could demand a change in behaviour from publically traded polluters, by (1) using their voting rights, (2) divesting their holdings, and (3) shorting the stock of polluters. Jana Partners, a large hedge fund, recently announced plans to launch an ESG activist fund this year. To get a sense of effectiveness, consider that a concerted plan by the largest asset managers to divest from the equity of the 20 largest publically traded polluters would likely result in a drop in the stock prices of these polluters, all else equal. As illustrated in Table 2, gains from a market-neutral short position (properly sized and disclosed) could be used by an impact fund to finance pro-environmental activities.
Table 2: Environmental Activism
Step 1: start with 30,000 public equities
Step 2: use negative environmental filters (e.g. energy use per $US of sales, CO2 emissions, environmental files) to identify the worst polluters
Step 3: use negative financial filters (e.g. low revenue growth, high leverage, high price-to-book, high P/E) to select, among the worst polluters, those that have a poor financial profile
Step 4: short their stock (against a market-long)
Step 5: use gains (extracted from the shareholders of weak polluters) to finance environmental clean-up work.
Sources:
* Ocean Fish Census:
https://www.worldatlas.com/articles/how-many-fish-are-there-in-the-ocean.html
* Hedge Fund ESG Activism
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