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Biological Growth models as modifications of the exponential

Exponential growth [1]

This is a key function, as most models in biology uses it as a backbone.

Logistic Model

Classical form

Logistic Model of population growth : [2]

Or autocompetition for the substrate

Logistic may be also an exponential growth linked to substrate use (two ODE) : [3].

See the phase diagram and the linear relation of a and S. The organisms are sharing the substrate ressource.

Or exponential with explicit braking

Explicit braking function in logistic model, in this case the role of the substrate consumption disappears, the equation includes explicitly a "braking funstion" replacing the second ODE. [4]

Properties

3 solutions differing by initial conditions : [5]

a=K is an "attractor": the model is converging toward a=K, this is an equilibrum point (stable) ; a=0 is also an (instable) equilibrum point, close to a=0 the logistic model is close to the exponential model. See also the way to superimpose the graphs of two models: [6]

A generalisation

The generalized logistic model : [7]

A simple 2 populations competing for the substrate

2 populations (logistic) and one substrate : [8]

Monod's Model

Classical

Monod's growth model of bacteria classical equation: [9]

Explicit braking function on the exponential growth

The braking function is an hyberbol : [10]

Monod's Model with acceleration and other complications

And now an acceleration at the beginning : [11]
With a delayed growth and mortality : [12]
Monod's Chemostat model : [13]
A complication certainly unrealistic but graphical output is interesting : [14]

Monod's relatives

The closely-related Contois'Model : [15]

Baranyi bacteria growth model

this is the model but with a braking function : [16]

Exponential growth with other braking functions

Allee's model : [17]
Monomolecular:[18]
Gompertz's Model :[19]
Growth of individuals and Gompertz's Model : [20]
Von Bertalanffy's model :[21]

Another approach :Simply Diffusing

Three compartiments 1 :[22]
Three compartiments 2 :[23]
Excretion of fibroin from sericigen glands :[24]
Production and secretion of bacterial toxins: [25] [26]

Interactions

Interactions of populations

Lotka-Volterra

Interaction between preys and predator (Lotka-Volterra) :[27]
Interaction between preys and predator with logistic growth :[28]
Bacteria and ameoba :[29]
Preys and predator but spatial effect 3 populations :[30]

Holling-Tanner

Holling-Tanner :[31]
Holling-Tanner with instability :[32]
Holling's Model : [33]
Substrate diffusion growth and grazing (unrealist) : [34]

Interactions of biology and chemistry

Nitrogen fixation in soil by bacteria : [35]

Interaction biology physics

Here are bacteria in food exposed to thermic variations : [36]

Epidemiology

SIR Model :[37]
Simplistic AIDS Model : [38]
A (better) model of AIDS epidemics :[39]
Hosts and parasites (Anderson and May 1981) : [40]
Coexistence of competing parasites (Hochberg and Holt 1990) : [41]
A model of Leptospirosis infection in an African rodent to determine risk to humans: seasonal fluctuations and the impact of rodent control. Acta Trop. 2006 Oct;99(2-3):218-25. [42]

Pharmacology and pharmarmacodynamy

A PK/PD for 2 nephrotoxic antibiotics : [43]
A PK-PD model of anticancer drug : [44]
A tumor growth is inhibited by an anticancer agent : [45]

Others

Sinus : a way to schematize regular variations : [46]
Lorentz chaotic model (not biologic) :[47]

Of course here is your sandbox for you to test directly your owns models (we provide a simple exponential as backbone)

DynW Sandbox is at [48]

Models Zoo