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A More Realistic Model of Isle Royale: Predator Prey Interactions

Andrew E Long
This model illustrates predator prey interactions using real-life data of wolf and moose populations on the Isle Royale.

We incorporate logistic growth into the moose dynamics, and we replace the death flow of the moose with a kill rate modeled from the kill rate data found on the Isle Royale website.

A decent match to the data is made with
Wolf Death Rate = 0.15
Wolf Birth Rate Factor = 0.0203
Moose Death Rate Factor = 1.08
Moose Birth Rate = 0.4
Carrying Capacity = 2000
Initial Moose: 563
Initial Wolves: 20

I used RK-4 with step-size 0.1, from 1959 for 60 years.

The moose birth flow is MBR*M*(1-M/K)
Moose death flow is MDRF*Sqrt(M*W)
Wolf birth flow is WBRF*Sqrt(M*W)
Wolf death flow is WDR*W

Environment Ecology Populations Midterm

  • 2 years 3 months ago

Isle Royale: Predator/Prey Model for Moose and Wolves, with Total Population

Andrew E Long
This model illustrates predator prey interactions using real-life data of wolf and moose populations on the Isle Royale. It was "cloned" from a model that InsightMaker provides to its users, at
Thanks Scott Fortmann-Roe.

I've added in an adjustment to handle population.

I've created a Mathematica file that replicates the model, at

It allows one to experiment with adjusting the initial number of moose and wolves on the island.

I used steepest descent in Mathematica to optimize the parameters, with my objective data being the ratio of wolves to moose. You can try my (admittedly) kludgy code, at

woStart} =

Environment Ecology Populations Math Modeling

  • 2 years 3 months ago

Sleek, non-dimensionalized Isle Royale: Predator/Prey Model for Moose and Wolves

Andrew E Long
This non-dimensionalized, sleekest most neatest model illustrates predator prey interactions using logistic growth for the moose population, for the wolf and moose populations on Isle Royale.

Thanks Scott Fortmann-Roe for the original model.

I've added in an adjustment to handle population sizes, by dividing by moose carrying capacity.

Time is scaled by the moose birth parameter:

There are therefore only three parameters left to account for any dynamics:

beta = bw/bm (relative wolf to moose births)
delta = dm/bm (relative death to birth ratio for moose)
gamma = dw/bm (wolf deaths to moose births)

The equations are thus

dM/dtau = M [ (1-M) - delta W ]
dW/dtau = W [beta M - gamma ]

There is a stable equilibrium pair of population values, relative to the carrying capacity:

M^* = gamma / beta
W^* = (1-gamma / beta) / delta

I have a sleek version with a logistical growth term for the moose, at

Environment Ecology Populations Math Modeling

  • 2 years 4 months ago

Population dynamics with overshoot ("Seneca cliff"?)

Barry McMullin
Start with logistic population dynamics (which can't overshoot) but then add delay in the "feedback signal" (the approach to the carrying capacity). One species, able to exploit one resource, which is available at  a fixed, finite, flow (not a depleting stock). At low populations, growth is exponential. As long as population below carrying capacity, growth continues. Without delay, it will smoothly stabilize at the carrying capacity. But with delay, it will overshoot; but oscillation should dampen, so eventually still stabilizes. Similar dynamics. "from above" (if, e.g., "initial" population somehow above carrying capacity; or, more plausibly, if carrying capacity dynamically falls to some lower level). With more delay, get more extreme overshoot. In "extreme" cases (relatively large delay, large overshoot) we can note asymmetry in "boom" and "bust" - bust is more rapid. This can be interpreted as a very simple version of Bardi's Seneca Cliff.


  • 11 months 2 weeks ago

Clone of Levels of transition needed to sustainability

Challenges in sustainability are multilevel.
This diagram attempts to summarize levels of self reinforcing destructive dynamics, authors that deal with them, and point of leverage for change.

The base of the crisis is a mechanistic rather than ecological worldview. This mechanistic worldview is based on outdated science that assumed the universe to be a large machine. In a machine there is an inside and an outside. The health of the inside is important for the machine, the outside not. In an ecological view everything is interconnected, there is no clear separation in the future of self and other. All parts influence the health of other parts. To retain health sensitivity and democracy are inherent. The sense of separation from other that keeps the mechanistic worldview dominant is duality. Being cut off from spiritual traditions due to a mechanistic view of science people need access to inter-spirituality to reconnect with the human traditions and tools around connectedness, inner discovery, and compassion. Many books on modern physics and biology deal with the system view implications. "The coming interspiritual age" deals with the need to connect spiritual traditions and science.

At the bottom for the dynamic is an individual a sense of disconnectedness leads to a dependency on spending and having rather than connecting. The connecting has become too painful and dealing with it unpopular in our culture. Joanna Macy deals with this in Active Hope. 

This affluenza and disconnection is worsened by a market that floods one with advertisements aimed at creating needs and a sense of dissatisfaction with that one has.

National economies are structured around maximising GDP which means maximising consumption and financial capital movement. This is at the cost of local economies. These same local economies are needed for balanced happiness as well as for sustainability.

Generally institutions focus on maximising consumption rather than sustaining life support systems. David Korten covers this well.

Power and wealth is confused in this worldview. In striving for wealth only power is striven for in the form of money and monopoly.

Those at the head of large banks and corporations tend to be there because they exemplify this approach. They have few scruples about enforcing this approach onto everyone through wars and disaster capitalism. Naomi Klein and David Estulin documented this.

Power has become so centralized that we need this understanding to be widespread and include many of those in power. Progress of all of these levels are needed to show them and all that another way is possible.

Environment Power Capitalism Exploitation Affluenza Sustainability Crisis Ecology Transition The Great Turning

  • 6 years 2 weeks ago

Testing of Caribou Conservation Sub-Models v2

Rob Rempel
Woodland caribou is a species at risk because of northward expansion of resource development activity.  Some herds are in dire condition and well below self-sustainability, while others are only moderately below self-sustaining levels.  Given limited conservation dollars, what are the most effective conservation actions, and how much money needs to be spent?  Which herds should be a priority for conservation efforts? The purpose of this model to provide insight into these difficult conservation questions.  
This model was developed by Rob Rempel and Jen Shuter at the Centre for Northern Forest Ecosystem Research, and was based in part on input from attendees of a modelling workshop ("Modelling the Caribou Questions") held at the 16th North American Caribou Workshop in Thunder Bay, Ontario, May 2016.

Population Caribou Wolves Moose Conservation Triage Ecology

  • 3 years 11 months ago


John Hearne
Under constructionTo understand the consequences of managing at a broad scale when divers are harvesting at a finer scale.


  • 6 years 4 weeks ago

Dystopia: simple energy system model

Barry McMullin
Basic idea is to model demand with endogenous growth (but "satiation" becomes possible - eventually - at some notional "sufficiency" level); and supply then attempts to track demand with some time lag (~5-50 years - characteristic of commissioning/decommissioning large scale energy infrastructure). But supply also produces pollution, which accoumulates. We can specify a notional constraint/limit; approaching this should trumpdemand and forces supply to zero. In this version we'll only have one source. so no substitution is possible. We expect to see a fairly sudden supply crash. Of course, "demand" will still carry merrily on its way up anyway, but the interpretation of the consequently growing supply shortfall will be left to the eye of the beholder. NB: this version doesn't automatically succeed in limiting P to P_max. It forces dS/dt to zero as A*(P/ P_max) reaches 1; and then as that value exceeds 1, dS/dt is forced negative. But this dynamics has no way to "undo" any overshoot of P over P_max (which would require S itself to become negative: "negative emissions"). Need to manual find/choose a big enough value of A to limit P effectively.

Energy Environment Ecology

  • 1 year 10 months ago