Wolfram Demonstrations Project offers free interactive visualisations and a free Mathematica Player. I have authored the following demonstrations:

The Gordon-Schaefer Model

The Gordon–Schaefer model is a bioeconomic comparative static fishery model based on logistic biological growth, constant harvest price, constant unit cost of effort, and harvest linear in stock biomass and fishing effort. TR denotes total revenue, TC total cost, AR average revenue (TR/E), MR marginal revenue, and MC marginal cost.

Surplus Production in Logistic Growth

The Verhulst growth function (red curve) consists of the logistic growth function (lower blue curve) and its time derivative.

Quasi-Rent in Open Access Fisheries

A phase plot (on the left) of an open access fishery dynamic system includes biological dynamics (blue isocline) and economic dynamics (red isocline). The first represents biological growth by a logistic growth equation while growth in fishing effort is proportional to obtained rent. Corresponding developments in effort and rent (rent adjustment, labeled quasi-rent by Alfred Marshall) as functions of time are shown on the right. The equilibrium solution gains no rent, while positive or negative rent is obtained outside equilibrium. Positive and negative quasi-rent are shown as functions of time in the lower panel on the right as green or red areas, respectively. The discounted sum constitutes the total quasi-rent presented.

Cellular Automata Model of an MPA Fishery

This model uses a continuous cellular automata technique to model the distribution of a fish stock along a coastline. The distribution of fishing activities is related to previously obtained rent from the fishery and initial fishing effort is controlled by a slider. Stock biomass growth depends on local density; local stock collapses occur when the stock biomass passes the environmental saturation level. All cells (localities) have the same growth properties. Biomass as a function of time is shown in the two panels on the left; the upper panel shows biomass gradients distributed over time and area (coastline), while the lower panel shows the total biomass as a function of time, the dashed line indicating the average natural stock biomass level. Total rent over the period is discounted and shown on the right as the quasi-rent of an open access fishery, only restricted by economic parameters and an MPA (marine protected area) regulation. The MPA regulation is a fixed fraction of the total coast length, controlled by a slider. The closed (protected) area is indicated by the red bar on the right in the upper-left panel, while the open area is indicated by the green bar.

Bioeconomics of a Discrete Ricker Model with Delayed Recruitment

The discrete Ricker population model with delayed recruitment is given by for a fish stock population. denotes the stock biomass at time and is the intrinsic growth rate. An open access fishery is constrained by the biological properties of the stock, the cost/price ratio (cost of producing fishing mortality and unit price of harvest), and the rate of fleet entry/exit to the fishery. The entry/exit dynamics are linear to the profitability of the fishery, with increasing fishing mortality when economic rent (resource rent) is positive and decreasing fishing mortality when rent is negative. Stable equilibria are found as limit cycles or stationary equilibrium; chaotic behavior may also exist. While an increasing growth rate destabilizes the system, increasing fishing mortality tends to stabilize it. The natural equilibrium or stable focus (when fishing mortality is zero) is shown by the intersection of the two dashed lines. In the case of positive fishing mortality, corresponding equilibria are found south-west of the intersection between the dashed lines.

Surplus Production Models and Equilibrium Harvest

Textbook bioeconomics (here considering fisheries) assumes logistic population growth and short term harvest production to be linear in stock biomass and fishing effort. The first assumption is expressed by the growth equation F(X)=rX(1-X/K) (red curve), X being the stock biomass and the parameters r and K the intrinsic growth rate and the environmental saturation level, respectively. F(X) is the per period of time surplus production. X=K is the natural equilibrium in the absence of fishing. The second assumption is the bilinear harvest equation h(E,X)=qEX (green line), E being the fishing effort, while the parameter q is known as the catchability coefficient. The fishing activity disturbs the natural equilibrium (X=K) and each level of fishing effort includes new equilibria determined by F(X)=h(E,X). This Demonstration illustrates how the collection of all existing equilibria found by varying values of E describes the equilibrium catch h(E), (blue curve). Depensation (decrease in marginal growth by decreasing population biomass) may give rise to two equilibrium points, one stable and the other unstable. The dashed blue curve indicates where unstable equilibrium points are found. Positive critical depensation levels are found when depensation causes negative biomass growth. The term depensation refers to a situation where decline in biomass is not compensated by increased per unit of biomass production, as in the logistic growth equation, where F(X)/X=r(1-X/K) is a down-sloping line for


Non-Renewable Resource Economics

The capital-theoretic approach to non-renewable resource economics—as described by Harold Hotelling—is to exhaust the resource over time while maximizing the present value of the resource (Hotelling's rule). This Demonstration illustrates the case of producer market power, indicated by the blue demand curve declining by increasing quantity (note that the quantity axis points to the left).

The total resource stock is represented by the shaded area in the lower-left panel enveloped by the green curve, which is the extraction rate. The shadowed area is a finite area, limited by the extraction start (initial extraction value given by the green curve at time zero) and the point of completed stock extraction (the green curve intersects the time axis where the extraction quantity reach zero). Depending on the chosen values, all parts of the finite area defined by the green curve may not be visible within the range of the axes.

The red curve is the optimal price of the resource owner according to the Hotelling’s rule, , where is the unit price of the resource and is the resource owner’s discount rate. In this Demonstration there are no extraction costs.

The shape of the green curve is determined by the stock size and discount rate while the red curve is determined by the discount rate and the demand for the extracted resource. The blue demand curve is in principle given exogenously, influencing the optimal exploitation path of the resource owner.

Drag the locator connecting the three curves to study the relation between them. The interconnection between the red, green and blue curves represents a point in time of optimal exploitation according to the given demand, stock size, and discount rate. The shaded area below the dashed line representing the point in time of optimal extraction is the remaining stock size at that very point in time. The capital-theoretic conclusion is that the optimal extraction path is found when the marginal growth in the value of the resource at every point in time equals the discount rate, as stated in the Hotelling's rule.

Offspring of Adam and Eve

This Demonstration starts with two people (Adam and Eve) at year 0. After some years they get their first child. Each year after that, they may get a new child or not, depending on the predefined number of children.

Observe the effect of using birth control (reducing the number of children) compared with increasing the age of first birth. You may be surprised!

Beverton and Holt's Yield per Recruit Model

In 1957 R. J. H. Beverton and S. J. Holt published their seminal work on the dynamics of exploited fish populations. The work includes individual growth models of von Bertalanffy, mortality models of Baranov, and a number of other methods to describe population dynamics. The most famous model presented in the work of Beverton and Holt is the simple yield per recruit model, presenting the sustainable yield of a fish population as a function of age of first catch tc, assuming knife-sharp selection and rate of fishing mortality F. The weight-length-relation b is assumed to be close to 3 (cubic relation) and the natural mortality rate M is in most cases close to the natural individual growth rate k.

Origami Bases

The art of origami involves some basic forms referred to as origami bases. The origami bases are certain sets of creases usually named after the most common final figure. This Demonstration presents 13 of the most common origami bases and how they should be folded.

Optimal Queue of Cars

This Demonstration considers the problem of finding the optimal speed in a queue of cars. It allows a sufficient distance between cars that depends on the road quality (due to friction) and the reaction time of the drivers and cars. Short, medium, or long cars carry 1, 1.5, or 2 people per meter of car.

A Nonlinear Stage-Structured Cannibalism Model

A cannibalism model published in [1] is described by a system of difference equations:


where and are the immature and mature parts of the population at time , respectively, (density independent) is the fecundity (number of newborns per adult), and is the fraction (density independent) of the immature population that survives and enters the mature stage one unit time later (

). The parameters and may be interpreted as the natural death rates (not the fishing mortality) caused by factors other than cannibalism. The nonlinear interactions in the model are of the Ricker type and the corresponding parameters

will be referred to as the cannibalism parameters. The model has a large parameter region with stable equilibrium points. It may transition from stable to unstable through a (supercritical) Hopf bifurcation, which means that beyond the instability threshold the dynamics are a quasiperiodic orbit restricted to an invariant curve that surrounds the unstable equilibrium

and ).

Continuous and Discrete Time Discounting

The well-known concept of discounting may be implemented as a discrete or continuous process in time, the first representing the common approach in financial institutions. The discrete time discounting term is

, where is the discount rate and is the time variable. The expression may be regarded as the present value of one unit of value at time . For , the expression decreases over time. The corresponding continuous time expression is

. Note that . The integral is shown as the PV (red) area (the present value of receiving one unit of value each unit of time eternally), while the PV (blue) area represents the sum. You can see the discrete time discounting as the light blue bars and/or a connecting blue line.

Word Memory Game

Word Memory is a simple game where the aim is to find a five-letter hidden word. For each try the player gets to know how many of the proposed letters are in the hidden word. If for example the hidden word is "faced" and your suggestion is "place", it gives 3 points (for the letters "a", "c" and "e"), independent of the position of letters. Only valid words (found in the dictionary) are accepted and a word may only be used once. If the same letter is found twice in the word (as the "d" in "dread") and you suggest another word with that letter also repeated, you receive two points.

In the game you challenge your computer and start the game by deciding the hidden word for the computer, before suggesting your own words. A table on the right-hand side lists all suggested words of the game, the red being your proposals and the blue the computer's. Average points of each player are displayed below the table.

Ecosystem Dynamics

Consider an ecosystem consisting of three trophic levels, 1 being the lowest and 3 the top predator level. Let the system be described by a set of differential equations, each representing the biomass dynamics of one of the three levels. The model is within the basic framework introduced by May and co-workers [1];

represents biomasses of the trophic level.




A biologically consistent system is obtained with non-negative parameter values. The corner solution reduces the system to just the lowest trophic level (1), while and

, , , and positive defines a system of three distinct levels, where levels 1 and 3 only interact through level 2.

Initial biomass levels are indicated in the graph by a red point, while corresponding terminal values are found at the green point. The connecting blue curve is the time path of the biomass development within the three trophic levels between these two points. The three bars give a graphical representation of the terminal biomasses of the three levels.

Guitar Chord Maker

Guitar chords are sets of tones created by pressing the strings against the frets on the guitar fingerboard. This Demonstration identifies different possible finger positions that create different kind of chords. A large variety of different chords are found in the advanced version.

Guitars may be tuned in different ways; this is determined here by the base tune of the sixth (thickest) string and the intervals between the strings.

Other demonstrations not yet published by Wolfram Research (flash previews not available):


webMathematica adds computations and visualisation to web sites. This are examples hosted by Norwegian College of Fishery Science, University of Tromsø: