From 6c9ec4f093ecfa48fe3a4a3fa99de16c5676d7dc Mon Sep 17 00:00:00 2001 From: bnewbold Date: Mon, 16 Jan 2017 16:24:09 -0800 Subject: update remaining models --- examples/hodgkin_huxley/model.modelica | 32 -------- examples/hodgkin_huxley/page.md | 86 ---------------------- .../hodgkin_huxley_neron_potential/model.modelica | 32 ++++++++ examples/hodgkin_huxley_neron_potential/page.md | 86 ++++++++++++++++++++++ examples/lotka_volterra/examples.toml | 11 --- examples/lotka_volterra/model.modelica | 12 --- examples/lotka_volterra/page.md | 35 --------- .../lotka_volterra_predator_prey/examples.toml | 11 +++ .../lotka_volterra_predator_prey/model.modelica | 12 +++ examples/lotka_volterra_predator_prey/page.md | 35 +++++++++ examples/van_der_pol/examples.toml | 0 examples/van_der_pol/model.modelica | 8 -- examples/van_der_pol/page.md | 0 examples/van_der_pol_oscillator/model.modelica | 8 ++ examples/van_der_pol_oscillator/page.md | 22 ++++++ examples/von_bertalanffy/model.modelica | 7 -- examples/von_bertalanffy/page.md | 5 -- .../von_bertalanffy_organism_growth/model.modelica | 8 ++ examples/von_bertalanffy_organism_growth/page.md | 22 ++++++ 19 files changed, 236 insertions(+), 196 deletions(-) delete mode 100644 examples/hodgkin_huxley/model.modelica delete mode 100644 examples/hodgkin_huxley/page.md create mode 100644 examples/hodgkin_huxley_neron_potential/model.modelica create mode 100644 examples/hodgkin_huxley_neron_potential/page.md delete mode 100644 examples/lotka_volterra/examples.toml delete mode 100644 examples/lotka_volterra/model.modelica delete mode 100644 examples/lotka_volterra/page.md create mode 100644 examples/lotka_volterra_predator_prey/examples.toml create mode 100644 examples/lotka_volterra_predator_prey/model.modelica create mode 100644 examples/lotka_volterra_predator_prey/page.md delete mode 100644 examples/van_der_pol/examples.toml delete mode 100644 examples/van_der_pol/model.modelica delete mode 100644 examples/van_der_pol/page.md create mode 100644 examples/van_der_pol_oscillator/model.modelica create mode 100644 examples/van_der_pol_oscillator/page.md delete mode 100644 examples/von_bertalanffy/model.modelica delete mode 100644 examples/von_bertalanffy/page.md create mode 100644 examples/von_bertalanffy_organism_growth/model.modelica create mode 100644 examples/von_bertalanffy_organism_growth/page.md diff --git a/examples/hodgkin_huxley/model.modelica b/examples/hodgkin_huxley/model.modelica deleted file mode 100644 index 7e44547..0000000 --- a/examples/hodgkin_huxley/model.modelica +++ /dev/null @@ -1,32 +0,0 @@ -model HodgkinHuxley - "Model of action potential in squid neurons (1952)" - parameter Real C_m =1.0 "membrane capacitance"; - parameter Real g_Na =120 "conductance"; - parameter Real g_K =36 "conductance"; - parameter Real g_L =0.3 "conductance"; - parameter Real V_Na =115 "potential"; - parameter Real V_K =-12 "potential"; - parameter Real V_lk =-49.387 "leak reveral potential"; - parameter Real E_Na =-190 "equilibrium potential"; - parameter Real E_K =-63 "equilibrium potential"; - parameter Real E_lk =-85.613 "equilibrium potential"; - parameter Real n =0.31768 "dimensionless; 0 to 1"; - parameter Real m =0.05293 "dimensionless; 0 to 1"; - parameter Real h =0.59612 "dimensionless; 0 to 1"; - Real V_m "membrane voltage potential"; - Real I =1.0 "membrane current"; - Real alpha_n, alpha_m, alpha_h "rate constants"; - Real beta_n, beta_m, beta_h "rate constants"; -equation - C_m * der(V_m) = I - g_Na * m^3 * h * (V_m - E_Na) - g_K * n^4 * (V_m - E_K) - G_lk * (V_m - E_lk); - der(n) = alpha_n - n * (alpha_n + beta_n); - der(m) = alpha_m - m * (alpha_m + beta_m); - der(h) = alpha_h - h * (alpha_h + beta_h); - - alpha_n = 0.01 * (V_m + 10) / (e^((V_m + 10)/10) - 1); - alpha_m = 0.1 * (V_m + 25) / (e^((V_m + 25)/10) - 1); - alpha_h = 0.07 * e^(V_m / 20); - beta_n = 0.125 * e^(V_m / 80); - beta_m = 4*e^(V_m/18); - beta_h = 1 / (e^((V_m + 30)/10) + 1); -end HodgkinHuxley; diff --git a/examples/hodgkin_huxley/page.md b/examples/hodgkin_huxley/page.md deleted file mode 100644 index 4598c97..0000000 --- a/examples/hodgkin_huxley/page.md +++ /dev/null @@ -1,86 +0,0 @@ - -The Hodgkin–Huxley model, or conductance-based model, is a mathematical model -that describes how action potentials in neurons are initiated and propagated. -It is a set of nonlinear differential equations that approximates the -electrical characteristics of excitable cells such as neurons and cardiac -myocytes, and hence it is a continuous time model, unlike the Rulkov map for -example. - -Alan Lloyd Hodgkin and Andrew Fielding Huxley described the model in 1952 to -explain the ionic mechanisms underlying the initiation and propagation of -action potentials in the squid giant axon. They received the 1963 Nobel Prize -in Physiology or Medicine for this work. - -## Mathematical properties - -The Hodgkin–Huxley model can be thought of as a differential equation with four -state variables, v(t), m(t), n(t), and h(t), that change with respect to time -t. The system is difficult to study because it is a nonlinear system and cannot -be solved analytically. However, there are many numeric methods available to -analyze the system. Certain properties and general behaviors, such as limit -cycles, can be proven to exist. - -## Alternative Models - -The Hodgkin–Huxley model is regarded as one of the great achievements of 20th-century biophysics. Nevertheless, modern Hodgkin–Huxley-type models have been extended in several important ways: - -* Additional ion channel populations have been incorporated based on experimental data. - -* The Hodgkin–Huxley model has been modified to incorporate transition state - theory and produce thermodynamic Hodgkin–Huxley models. - -* Models often incorporate highly complex geometries of dendrites and axons, - often based on microscopy data. - -* Stochastic models of ion-channel behavior, leading to stochastic hybrid - systems - -Several simplified neuronal models have also been developed (such as the -FitzHugh–Nagumo model), facilitating efficient large-scale simulation of groups -of neurons, as well as mathematical insight into dynamics of action potential -generation. - - -## References - -The body of this page is from Wikipedia (see below). - -#### Papers - -"The dual effect of membrane potential on sodium conductance in the giant axon -of Loligo". *The Journal of Physiology*. **116** (4): 497–506. April 1952. -doi:10.1113/jphysiol.1952.sp004719. - -"Currents carried by sodium and potassium ions through the membrane of the -giant axon of Loligo". *The Journal of Physiology*. **116** (4): 449–72. April 1952. -doi:10.1113/jphysiol.1952.sp004717. - -"The components of membrane conductance in the giant axon of Loligo". *The -Journal of Physiology*. **116** (4): 473–96. April 1952. -doi:10.1113/jphysiol.1952.sp004718. - -"The dual effect of membrane potential on sodium conductance in the giant axon -of Loligo". *The Journal of Physiology*. **116** (4): 497–506. April 1952. -doi:10.1113/jphysiol.1952.sp004719. - -"A quantitative description of membrane current and its application to -conduction and excitation in nerve". *The Journal of Physiology*. **117** (4): -500–44. August 1952. doi:10.1113/jphysiol.1952.sp004764. - -#### Interactive Models on the Web - -* ModelDB: [Squid axon (Hodgkin, Huxley 1952)](https://senselab.med.yale.edu/ModelDB/ShowModel.cshtml?model=5426) -* Wolfram Demonstrations: - [Interactive Hodgkin-Huxley](http://demonstrations.wolfram.com/HodgkinHuxleyActionPotentialModel/) - by Shimon Marom and - [Neural Impulses: The Action Potential in Action](http://www.demonstrations.wolfram.com/NeuralImpulsesTheActionPotentialInAction/) - by Garrett Neske -* [Hodgkin-Huxley Simulation with Javascript](http://myselph.de/hodgkinHuxley.html) - by Hubert Eichner, which creates static plots in the browser. -* BioModels database: [](http://www.ebi.ac.uk/biomodels-main/BIOMD0000000020) - -#### Other Links - -* Wikipedia: [Hodgkin–Huxley model](https://en.wikipedia.org/wiki/Hodgkin%E2%80%93Huxley_model) -* [Summary of the Hodgkin-Huxley model](http://ecee.colorado.edu/~ecen4831/HHsumWWW/HHsum.html) -* [Hodgkin-Huxley model in R](http://www.magesblog.com/2012/06/hodgkin-huxley-model-in-r.html) diff --git a/examples/hodgkin_huxley_neron_potential/model.modelica b/examples/hodgkin_huxley_neron_potential/model.modelica new file mode 100644 index 0000000..7e44547 --- /dev/null +++ b/examples/hodgkin_huxley_neron_potential/model.modelica @@ -0,0 +1,32 @@ +model HodgkinHuxley + "Model of action potential in squid neurons (1952)" + parameter Real C_m =1.0 "membrane capacitance"; + parameter Real g_Na =120 "conductance"; + parameter Real g_K =36 "conductance"; + parameter Real g_L =0.3 "conductance"; + parameter Real V_Na =115 "potential"; + parameter Real V_K =-12 "potential"; + parameter Real V_lk =-49.387 "leak reveral potential"; + parameter Real E_Na =-190 "equilibrium potential"; + parameter Real E_K =-63 "equilibrium potential"; + parameter Real E_lk =-85.613 "equilibrium potential"; + parameter Real n =0.31768 "dimensionless; 0 to 1"; + parameter Real m =0.05293 "dimensionless; 0 to 1"; + parameter Real h =0.59612 "dimensionless; 0 to 1"; + Real V_m "membrane voltage potential"; + Real I =1.0 "membrane current"; + Real alpha_n, alpha_m, alpha_h "rate constants"; + Real beta_n, beta_m, beta_h "rate constants"; +equation + C_m * der(V_m) = I - g_Na * m^3 * h * (V_m - E_Na) - g_K * n^4 * (V_m - E_K) - G_lk * (V_m - E_lk); + der(n) = alpha_n - n * (alpha_n + beta_n); + der(m) = alpha_m - m * (alpha_m + beta_m); + der(h) = alpha_h - h * (alpha_h + beta_h); + + alpha_n = 0.01 * (V_m + 10) / (e^((V_m + 10)/10) - 1); + alpha_m = 0.1 * (V_m + 25) / (e^((V_m + 25)/10) - 1); + alpha_h = 0.07 * e^(V_m / 20); + beta_n = 0.125 * e^(V_m / 80); + beta_m = 4*e^(V_m/18); + beta_h = 1 / (e^((V_m + 30)/10) + 1); +end HodgkinHuxley; diff --git a/examples/hodgkin_huxley_neron_potential/page.md b/examples/hodgkin_huxley_neron_potential/page.md new file mode 100644 index 0000000..4598c97 --- /dev/null +++ b/examples/hodgkin_huxley_neron_potential/page.md @@ -0,0 +1,86 @@ + +The Hodgkin–Huxley model, or conductance-based model, is a mathematical model +that describes how action potentials in neurons are initiated and propagated. +It is a set of nonlinear differential equations that approximates the +electrical characteristics of excitable cells such as neurons and cardiac +myocytes, and hence it is a continuous time model, unlike the Rulkov map for +example. + +Alan Lloyd Hodgkin and Andrew Fielding Huxley described the model in 1952 to +explain the ionic mechanisms underlying the initiation and propagation of +action potentials in the squid giant axon. They received the 1963 Nobel Prize +in Physiology or Medicine for this work. + +## Mathematical properties + +The Hodgkin–Huxley model can be thought of as a differential equation with four +state variables, v(t), m(t), n(t), and h(t), that change with respect to time +t. The system is difficult to study because it is a nonlinear system and cannot +be solved analytically. However, there are many numeric methods available to +analyze the system. Certain properties and general behaviors, such as limit +cycles, can be proven to exist. + +## Alternative Models + +The Hodgkin–Huxley model is regarded as one of the great achievements of 20th-century biophysics. Nevertheless, modern Hodgkin–Huxley-type models have been extended in several important ways: + +* Additional ion channel populations have been incorporated based on experimental data. + +* The Hodgkin–Huxley model has been modified to incorporate transition state + theory and produce thermodynamic Hodgkin–Huxley models. + +* Models often incorporate highly complex geometries of dendrites and axons, + often based on microscopy data. + +* Stochastic models of ion-channel behavior, leading to stochastic hybrid + systems + +Several simplified neuronal models have also been developed (such as the +FitzHugh–Nagumo model), facilitating efficient large-scale simulation of groups +of neurons, as well as mathematical insight into dynamics of action potential +generation. + + +## References + +The body of this page is from Wikipedia (see below). + +#### Papers + +"The dual effect of membrane potential on sodium conductance in the giant axon +of Loligo". *The Journal of Physiology*. **116** (4): 497–506. April 1952. +doi:10.1113/jphysiol.1952.sp004719. + +"Currents carried by sodium and potassium ions through the membrane of the +giant axon of Loligo". *The Journal of Physiology*. **116** (4): 449–72. April 1952. +doi:10.1113/jphysiol.1952.sp004717. + +"The components of membrane conductance in the giant axon of Loligo". *The +Journal of Physiology*. **116** (4): 473–96. April 1952. +doi:10.1113/jphysiol.1952.sp004718. + +"The dual effect of membrane potential on sodium conductance in the giant axon +of Loligo". *The Journal of Physiology*. **116** (4): 497–506. April 1952. +doi:10.1113/jphysiol.1952.sp004719. + +"A quantitative description of membrane current and its application to +conduction and excitation in nerve". *The Journal of Physiology*. **117** (4): +500–44. August 1952. doi:10.1113/jphysiol.1952.sp004764. + +#### Interactive Models on the Web + +* ModelDB: [Squid axon (Hodgkin, Huxley 1952)](https://senselab.med.yale.edu/ModelDB/ShowModel.cshtml?model=5426) +* Wolfram Demonstrations: + [Interactive Hodgkin-Huxley](http://demonstrations.wolfram.com/HodgkinHuxleyActionPotentialModel/) + by Shimon Marom and + [Neural Impulses: The Action Potential in Action](http://www.demonstrations.wolfram.com/NeuralImpulsesTheActionPotentialInAction/) + by Garrett Neske +* [Hodgkin-Huxley Simulation with Javascript](http://myselph.de/hodgkinHuxley.html) + by Hubert Eichner, which creates static plots in the browser. +* BioModels database: [](http://www.ebi.ac.uk/biomodels-main/BIOMD0000000020) + +#### Other Links + +* Wikipedia: [Hodgkin–Huxley model](https://en.wikipedia.org/wiki/Hodgkin%E2%80%93Huxley_model) +* [Summary of the Hodgkin-Huxley model](http://ecee.colorado.edu/~ecen4831/HHsumWWW/HHsum.html) +* [Hodgkin-Huxley model in R](http://www.magesblog.com/2012/06/hodgkin-huxley-model-in-r.html) diff --git a/examples/lotka_volterra/examples.toml b/examples/lotka_volterra/examples.toml deleted file mode 100644 index 0f5fb85..0000000 --- a/examples/lotka_volterra/examples.toml +++ /dev/null @@ -1,11 +0,0 @@ - -[examples] - - [examples.mathworld] - x = 10 - y = 5 - alpha = 1.5 - beta = 1 - gamma = 1 - t = [0, 20] - diff --git a/examples/lotka_volterra/model.modelica b/examples/lotka_volterra/model.modelica deleted file mode 100644 index 706ebc3..0000000 --- a/examples/lotka_volterra/model.modelica +++ /dev/null @@ -1,12 +0,0 @@ -model LotkaVolterra - "Predator-Prey equations describing population dynamics of two biological species" - parameter Real alpha; - parameter Real beta; - parameter Real delta; - parameter Real gamma; - Real x "population of prey"; - Real y "population of predator"; -equation - der(x) = (alpha * x) - (beta * x * y); - der(y) = (delta * x * y) - (gamma * y); -end LotkaVolterra; diff --git a/examples/lotka_volterra/page.md b/examples/lotka_volterra/page.md deleted file mode 100644 index 1ae891d..0000000 --- a/examples/lotka_volterra/page.md +++ /dev/null @@ -1,35 +0,0 @@ - -The Lotka–Volterra equations, also known as the predator–prey equations, are a -pair of first-order, [non-linear](https://en.wikipedia.org/wiki/Non-linear), -differential equations frequently used to describe the dynamics of biological -systems in which two species interact, one as a predator and the other as prey. - -The Lotka–Volterra system of equations is an example of a Kolmogorov -model, which is a more general framework that can model the dynamics of -ecological systems with predator-prey interactions, competition, disease, and -mutualism. - -## Solutions to the equations - -The equations have periodic solutions and do not have a simple expression in -terms of the usual trigonometric functions, although they are quite -tractable. - -If none of the non-negative parameters α,β,γ,δ vanishes, three can be absorbed -into the normalization of variables to leave but merely one behind: Since the -first equation is homogeneous in x, and the second one in y, the parameters β/α -and δ/γ, are absorbable in the normalizations of y and x, respectively, and γ -into the normalization of t, so that only α/γ remains arbitrary. It is the only -parameter affecting the nature of the solutions. - -A linearization of the equations yields a solution similar to simple harmonic -motion with the population of predators trailing that of prey by 90° in the -cycle. - -## References - -Body text taken from Wikipedia. - -* [Mathworld](http://mathworld.wolfram.com/Lotka-VolterraEquations.html) -* [Wikipedia](https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations) -* [Population dynamics of fisheries](https://en.wikipedia.org/wiki/Population_dynamics_of_fisheries) diff --git a/examples/lotka_volterra_predator_prey/examples.toml b/examples/lotka_volterra_predator_prey/examples.toml new file mode 100644 index 0000000..0f5fb85 --- /dev/null +++ b/examples/lotka_volterra_predator_prey/examples.toml @@ -0,0 +1,11 @@ + +[examples] + + [examples.mathworld] + x = 10 + y = 5 + alpha = 1.5 + beta = 1 + gamma = 1 + t = [0, 20] + diff --git a/examples/lotka_volterra_predator_prey/model.modelica b/examples/lotka_volterra_predator_prey/model.modelica new file mode 100644 index 0000000..706ebc3 --- /dev/null +++ b/examples/lotka_volterra_predator_prey/model.modelica @@ -0,0 +1,12 @@ +model LotkaVolterra + "Predator-Prey equations describing population dynamics of two biological species" + parameter Real alpha; + parameter Real beta; + parameter Real delta; + parameter Real gamma; + Real x "population of prey"; + Real y "population of predator"; +equation + der(x) = (alpha * x) - (beta * x * y); + der(y) = (delta * x * y) - (gamma * y); +end LotkaVolterra; diff --git a/examples/lotka_volterra_predator_prey/page.md b/examples/lotka_volterra_predator_prey/page.md new file mode 100644 index 0000000..1ae891d --- /dev/null +++ b/examples/lotka_volterra_predator_prey/page.md @@ -0,0 +1,35 @@ + +The Lotka–Volterra equations, also known as the predator–prey equations, are a +pair of first-order, [non-linear](https://en.wikipedia.org/wiki/Non-linear), +differential equations frequently used to describe the dynamics of biological +systems in which two species interact, one as a predator and the other as prey. + +The Lotka–Volterra system of equations is an example of a Kolmogorov +model, which is a more general framework that can model the dynamics of +ecological systems with predator-prey interactions, competition, disease, and +mutualism. + +## Solutions to the equations + +The equations have periodic solutions and do not have a simple expression in +terms of the usual trigonometric functions, although they are quite +tractable. + +If none of the non-negative parameters α,β,γ,δ vanishes, three can be absorbed +into the normalization of variables to leave but merely one behind: Since the +first equation is homogeneous in x, and the second one in y, the parameters β/α +and δ/γ, are absorbable in the normalizations of y and x, respectively, and γ +into the normalization of t, so that only α/γ remains arbitrary. It is the only +parameter affecting the nature of the solutions. + +A linearization of the equations yields a solution similar to simple harmonic +motion with the population of predators trailing that of prey by 90° in the +cycle. + +## References + +Body text taken from Wikipedia. + +* [Mathworld](http://mathworld.wolfram.com/Lotka-VolterraEquations.html) +* [Wikipedia](https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations) +* [Population dynamics of fisheries](https://en.wikipedia.org/wiki/Population_dynamics_of_fisheries) diff --git a/examples/van_der_pol/examples.toml b/examples/van_der_pol/examples.toml deleted file mode 100644 index e69de29..0000000 diff --git a/examples/van_der_pol/model.modelica b/examples/van_der_pol/model.modelica deleted file mode 100644 index c1aa99b..0000000 --- a/examples/van_der_pol/model.modelica +++ /dev/null @@ -1,8 +0,0 @@ -model VanDerPolOscillator - parameter Real mu "dampening strength"; - Real x; - Real y; -equation - der(x) = y; - der(y) = mu * y * (1 - x^2) - x; -end VanDerPolOscillator; diff --git a/examples/van_der_pol/page.md b/examples/van_der_pol/page.md deleted file mode 100644 index e69de29..0000000 diff --git a/examples/van_der_pol_oscillator/model.modelica b/examples/van_der_pol_oscillator/model.modelica new file mode 100644 index 0000000..c1aa99b --- /dev/null +++ b/examples/van_der_pol_oscillator/model.modelica @@ -0,0 +1,8 @@ +model VanDerPolOscillator + parameter Real mu "dampening strength"; + Real x; + Real y; +equation + der(x) = y; + der(y) = mu * y * (1 - x^2) - x; +end VanDerPolOscillator; diff --git a/examples/van_der_pol_oscillator/page.md b/examples/van_der_pol_oscillator/page.md new file mode 100644 index 0000000..ff24c7c --- /dev/null +++ b/examples/van_der_pol_oscillator/page.md @@ -0,0 +1,22 @@ + +The Van der Pol oscillator was originally proposed by the Dutch electrical +engineer and physicist Balthasar van der Pol while he was working at +Philips. Van der Pol found stable oscillations, which he subsequently +called relaxation-oscillations and are now known as a type of limit cycle in +electrical circuits employing vacuum tubes. When these circuits were driven +near the limit cycle, they become entrained, i.e. the driving signal pulls the +current along with it. Van der Pol and his colleague, van der Mark, reported in +the September 1927 issue of Nature that at certain drive frequencies an +irregular noise was deterministic chaos. + +The Van der Pol equation has a long history of being used in both the physical +and biological sciences. For instance, in biology, Fitzhugh and Nagumo +extended the equation in a planar field as a model for action potentials of +neurons. The equation has also been utilised in seismology to model the two +plates in a geological fault, and in studies of phonation to model the right +and left vocal fold oscillators. + +## References + +* Wikipedia: + [Van der Pol oscillator](https://en.wikipedia.org/wiki/Van_der_Pol_oscillator) diff --git a/examples/von_bertalanffy/model.modelica b/examples/von_bertalanffy/model.modelica deleted file mode 100644 index 4eb45f6..0000000 --- a/examples/von_bertalanffy/model.modelica +++ /dev/null @@ -1,7 +0,0 @@ -model VonBertalanffyGrowth - parameter Real K "growth rate"; - parameter Real L "asymptotic final size"; - Real l "length"; -equation - der(l) = K * (L - l); -end VonBertalanffyGrowth; diff --git a/examples/von_bertalanffy/page.md b/examples/von_bertalanffy/page.md deleted file mode 100644 index ea6e9f1..0000000 --- a/examples/von_bertalanffy/page.md +++ /dev/null @@ -1,5 +0,0 @@ - - -An alternative/solve variant is: - - l = L * (1 - e^(-K * t)) diff --git a/examples/von_bertalanffy_organism_growth/model.modelica b/examples/von_bertalanffy_organism_growth/model.modelica new file mode 100644 index 0000000..afc6f7f --- /dev/null +++ b/examples/von_bertalanffy_organism_growth/model.modelica @@ -0,0 +1,8 @@ +model VonBertalanffyGrowth + "Rate-of-growth model for biological organisms" + parameter Real K "growth rate"; + parameter Real L "asymptotic final length (L infinity)"; + Real l "length"; +equation + der(l) = K * (L - l); +end VonBertalanffyGrowth; diff --git a/examples/von_bertalanffy_organism_growth/page.md b/examples/von_bertalanffy_organism_growth/page.md new file mode 100644 index 0000000..79a643f --- /dev/null +++ b/examples/von_bertalanffy_organism_growth/page.md @@ -0,0 +1,22 @@ + +The Bertalanffy equation is an equation that describes the rate of growth of a +biological organism. The equation was offered by Ludwig von Bertalanffy in 1969. + +## Solution + +Integrating the equation gives: + + $$ l = L \(1 - e^{-Kt}\) $$ + +## References + +The body of this page comes from Wikipedia. + +* Wikipedia: + [Bertalanffy equation](https://en.wikipedia.org/wiki/Ludwig_von_Bertalanffy#Bertalanffy_equation) +* [The von Bertalanffy growth equation](http://www.pisces-conservation.com/growthhelp/index.html?von_bertalanffy.htm) + +### Papers + +Bertalanffy, L. von, (1969). *General System Theory*. New York: George +Braziller, pp. 136 -- cgit v1.2.3