Poisoned growth

Poisoned growth refers to a mathematical model for different growth processes of systems in which the magnification of stocks (such as the propagation of a population ) by an inhibitor (also inhibitor called, such as a toxin ) is braked. Ultimately, the size approaches zero (the population dies out). A distinction is made between externally poisoned growth , which can be used to describe systems in which the inhibitor is added from outside, and self-poisoned growth , which can be used to describe systems in which the production of the inhibitor depends on the size of the population.

properties

Model description

The growth model assumes a closed system , i. H. the inhibitors are not removed or broken down. In contrast to exponential growth , the growth factor is not constant , but represents a function of time. With this model, not only the growth process of the population can be considered, but also that of the inhibitor separately.

There is no inhibitor at the start time . Before the poison is added or released, the population therefore grows exponentially without restraint. The death rate here is practically zero. The effect of the poison depends on the amount of poison added and the poisoning factor as a measure of the specific degree of poisoning . ${\displaystyle t=0}$

The increasing intoxication slows down the growth process, with the population size initially increasing monotonously . With a certain amount of the inhibitor, the birth and death rates are the same. At this point, the growth rate is zero. The stock reaches its maximum ( high point ) here, which can be determined by means of differential calculation.

From that point on, the death rate exceeds the birth rate, so that the population shrinks or the population falls monotonically. The growth rate is now negative and decreases roughly in proportion to the size of the population and the amount of poison.

From a mathematical point of view, the population does not completely disappear because the x-axis forms the asymptote of the growth function. In the application, however, population sizes are mostly whole numbers , which is why different, very small values ​​no longer have any meaning and a complete extinction is assumed if the system is to follow such a course.

Modeling

The steady (or continuous) growth model is described by a differential equation (DGL). The DGL is solved using the “ variable separation ” method . The special solution of the DGL forms the explicit representation of the growth model and specifies the concrete growth function .

The discrete model of poisoned growth can be described by a recursive representation using a sequence derived from differences . Here, says the time difference an equidistant sequence of time points and the corresponding inventory sizes. Mathematically, a distinction is also made between exact and nourished discretization. The latter results here by using the explicit Euler method . A series expansion of the exponential function shows that both representations agree except for terms higher than 1st order. ${\displaystyle \Delta t}$ ${\displaystyle (t_{n})=(t_{0},t_{1},t_{2},\dotsc )}$ ${\displaystyle (B_{n})=(B_{0},B_{1},B_{2},\dotsc )}$

Essential terms and notation

• ${\displaystyle t}$ denotes time.
• ${\displaystyle B\left(t\right)}$be the size of the population under consideration .
• ${\displaystyle B\left(0\right)={{B}_{0}}}$identifies the initial balance ( initial condition ) at the point in time . ${\displaystyle t=0}$
• ${\displaystyle k}$be the species-specific growth constant of the population. It is a measure of the strength of growth and essentially describes the birth rate . ${\displaystyle \left(k>0\right)}$
• ${\displaystyle {B}'\left(t\right)}$ indicates the rate of growth.
• ${\displaystyle c}$be the poisoning constant, which indicates the toxic effect of the inhibitor (poison) on the population as a poison or drug-specific parameter . It essentially describes the death rate . ${\displaystyle \left(c>0\right)}$

Model of Poisoned Growth

Over time, a certain amount of a toxic inhibitor is added to the stand from the outside. The following describes the case in which the amount of poison increases linearly , i.e. proportionally to time. The effective growth factor decreases accordingly over time. ${\displaystyle (k-c\cdot t)}$

Differential equation:

${\displaystyle {B}'\left(t\right)={\frac {\operatorname {d} B}{\operatorname {d} t}}=\left(k-c\cdot t\right)B\left(t\right)}$

Explicit representation (growth function):

${\displaystyle B\left(t\right)=B\left(0\right){{\operatorname {e} }^{k\cdot t-{\frac {1}{2}}c\cdot {{t}^{2}}}}}$

Growth rate:

${\displaystyle {B}'\left(t\right)={\frac {\operatorname {d} B}{\operatorname {d} t}}=\left(k-c\cdot t\right)B\left(0\right){{\operatorname {e} }^{k\cdot t-{\frac {1}{2}}c\cdot {{t}^{2}}}}}$

Maximum of the growth function:

${\displaystyle {{B}_{\max }}=B\left(0\right)\cdot {{\operatorname {e} }^{\frac {{k}^{2}}{2c}}}}$ at ${\displaystyle t_{max}={\frac {k}{c}}}$

Exact, recursive representation:

${\displaystyle {{B}_{n+1}}={{B}_{n}}{{\operatorname {e} }^{\left(k-c\left(n+{\frac {1}{2}}\right)\Delta t\right)\Delta t}}}$

Approximate, recursive representation:

${\displaystyle {{B}_{n+1}}=\left(1+\Delta t\left(k-c\cdot n\Delta t\right)\right){{B}_{n}}}$

Models of Self-Poisoned Growth

With self- poisoned growth (also called growth with self-poisoning ), the population itself produces toxins during the growth process - mostly in the form of metabolic products that influence growth. There are different approaches to describe this process in the literature.

One equation model

First, a model is considered in which poison production is estimated from the initial phase of the growth and intoxication process. In this first phase one can assume that the amount of poison will increase proportionally to the initially exponentially increasing population and will not yet be influenced by the death of the population. Ultimately, the steadily increasing amount of poison will lead to the population becoming extinct. But then - contrary to the original assumption - no more poison is produced either. This model can be described by a differential equation with a time-dependent growth factor (as in externally poisoned growth). The growth function can be determined exactly from the closed solution.

Differential equation:

${\displaystyle B'(t)={\frac {\operatorname {d} B}{\operatorname {d} t}}=\left(k-c\cdot \operatorname {e} ^{k\cdot t}\right)B(t)}$

Explicit representation (growth function):

${\displaystyle B(t)=B(0){\operatorname {e} ^{k\cdot t-{\frac {c}{k}}\left({\operatorname {e} ^{kt}}-1\right)}}}$

Growth rate:

${\displaystyle B'(t)={\frac {\operatorname {d} B}{\operatorname {d} t}}=\left(k-c\cdot {\operatorname {e} ^{k\cdot t}}\right)B(0){\operatorname {e} ^{k\cdot t-{\frac {c}{k}}\left({\operatorname {e} ^{kt}}-1\right)}}}$

Maximum of the growth function:

${\displaystyle B_{\text{max}}=B(0){\frac {k}{c}}{\operatorname {e} ^{\left({\frac {c}{k}}-1\right)}}}$ at ${\displaystyle t_{\text{max}}={\frac {1}{k}}\ln {\frac {k}{c}}}$

Exact recursive representation:

${\displaystyle B_{n+1}=B_{n}{\operatorname {e} ^{\left(k\Delta t\,-\,{\frac {c}{k}}{\operatorname {e} ^{kn\Delta t}}\left(\operatorname {e} ^{k\Delta t}-1\right)\right)}}}$

Approximate recursive representation:

${\displaystyle B_{n+1}=\left(1+\Delta t\left(k-c\cdot {\operatorname {e} ^{kn\Delta t}}\right)\right)B_{n}}$

Two equation model

An alternative modeling of self-poisoned growth is obtained by describing both the population and the amount of poison by two equations. This model is related to the so-called predator-prey models . Here the temporal increase in the amount of poison (predator) is determined by the current population (prey).

Although the two differential equations could be converted into a single 2nd order equation, this is not considered further. As in the other cases, the Euler forward method can also be used here for numerical solution.

Differential equations:

${\displaystyle {\begin{aligned}&B'(t)={\frac {\operatorname {d} B}{\operatorname {d} t}}=\left(k-c\cdot G(t)\right)B(t)\\&G'(t)={\frac {\operatorname {d} G}{\operatorname {d} t}}=b\cdot B(t)\\\end{aligned}}}$

The states of equilibrium are exactly the states with . ${\displaystyle B=0}$

Approximate, recursive representation:

${\displaystyle {\begin{aligned}&B_{n+1}=\left(1+\Delta t\left(k-c\cdot G_{n}\right)\right)B_{n}\\&G_{n+1}=G_{n}+\Delta t\cdot b\cdot B_{n}\\\end{aligned}}}$

Examples

The following empirical processes can, to a certain extent, be described by poisoned growth.

For foreign-poisoned growth

• Pharmacokinetics : Toxic effects of drugs on pathogens

To inhibit the growth of bacteria, an antibiotic that is toxic to the bacteria is fed to a living being at intervals . This reduces the growth rate of the bacteria and thus their population size, so that the bacteria practically disappear.

• Ecology : addition of environmentally hazardous substances

Environmental toxins lead to changes within a biotope up to the extinction of individual species within a habitat or a biocenosis . This also includes the waste problem, forest death, over-fertilization and wastewater pollution.

For self-poisoned growth

• Alcoholic fermentation : fermentation of sugar-containing solutions by yeast cultures

When brewing beer z. B. when breathing the glycose alcohol as a waste product, which is toxic for the yeast cells, inhibits their reproduction and leads to their extinction.

• Biology : processes in a closed habitat

If, for example, water fleas are not supplied with fresh water in an aquarium despite sufficient food, they will die after initial reproduction as a result of poisoning from their own metabolic residues.

Individual evidence

1. ↑ Joachim Engel: Application-oriented mathematics: From data to function . An introduction to mathematical modeling for student teachers. Springer Verlag , Heidelberg 2010, ISBN 978-3-540-89086-7 , p. 203-205 .
2. ↑ Klaus Pommerening: Computer simulation of dynamic systems illustrated using the example of predator-prey systems and other growth models from ecology . Script for the internship in software engineering. Mainz 1987, p. 9-10 . [1] (PDF; 228 kB) accessed online on March 3, 2013
3. ↑ Joachim Engel: Application-oriented mathematics: From data to function . An introduction to mathematical modeling for student teachers. Springer Verlag , Heidelberg 2010, ISBN 978-3-540-89086-7 , p. 202 .
4. ↑ Growth functions  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. accessed online on February 12, 2013.  
5. ↑ Joachim Engel: Application-oriented mathematics: From data to function . An introduction to mathematical modeling for student teachers. Springer Verlag , Heidelberg 2010, ISBN 978-3-540-89086-7 , p. 203 .
6. ↑ Development of a population , p. 3 accessed online on February 13, 2013.

literature

• Joachim Engel: Application-Oriented Mathematics: From Data to Function . An introduction to mathematical modeling for student teachers. Springer Verlag , Heidelberg 2010, ISBN 978-3-540-89086-7 , p. 201-207 .
• Klaus Schilling (Hrsg.): Collection of formulas: Core curriculum mathematics Lower Saxony: Vocational high school . Eins Verlag, Cologne 2012, ISBN 978-3-427-07770-1 , pp. 42 .
• Dietmar Schoh, Thomas Jahnke (Hrsg.): Focus on mathematics: Upper secondary school in Bavaria, 12th grade . Cornelsen Verlag , Berlin 2010, ISBN 978-3-06-009152-2 , pp. 183-185 .