# Andre modeller

In this lecture we discuss alternative bioeconomic modelling approaches along the following lines:

• Modifications/extensions of the Gordon-Schaefer model, including the use of other surplus production models
• Cohort models
• Multispecies modelling

Modifications/extensions of the Gordon-Schaeffer model

A bioeconomic model is made up of the following components: A biological growth model, a harvest production function including costs of input factors and a revenue function.

### Competition

One dominant

The competitive exclusion principle proposed by Gause. The equilibrium defined by the intersection of the two isoclines is not stable, while the two equilibriums where each isocline intersect its own axis are stable.

The classical Gordon- Schaefer model includes a logistic biomass growth function, a short term catch equation linear in effort and stock biomass, constant unit cost of effort and constant unit price of harvest.

The most typical modifications are

• to exchange the biological growth functions by other surplus production functions. You may also obtain more information about other surplus production models here.
• to use a Cobb-Douglas production function to model short term harvest production
• to include harvest quantity dependent prices (representing some degrees of market power)
• to assume intra marginal rent deriving from marginal cost differences between fishing units

Cohort models

You may investigate the Mathematica demonstration of the well-known Beverton and Holt model by clicking at the figure. This demonstration includes however not any economic module. Beverton and Holt believed the envelope curve of the yield curves of varying tc-values (tc: Age of first catch) could be used as an connection to market economy. This envelope curve (called the eumetric curve) has however some specific properties which makes it less useful (or even useless) as the biological part of a bioeconomic model.

No dominant

This shows a competitive model with a stable interior solution.

Other representations of this cohort model:

The three links above present webMathematica pages where it is possible to calculate (Evaluate) graphs and equations (including analytical equations) of different inputs.

## Different dynamic systems are presented, all expressed by two differential equations. Equilibriums are shown in phase plots as the intersections between the two isoclines of the system (blue lines). The phase plots also display vector fields representing the dynamics of each system.

### Open access

Open access dynamics are well known and could be expressed by the two differential equations below:

### Symbiosis

Two species benefiting from each other.

A stable solution is found.

### Predator - Prey

Lotka - Volterra

A famous predator-prey relationship was expressed by Lotka and Volterra in 1925/1926. The system results in stable cycles (limited cycle), which orbit depends on the intial conditions.

You may also download a live version of Predator-Prey equations from Wolfram Demonstration Project.

Normal Predator - Prey

A more general expression, opening for stable equilibrium solutions.