Population logistic model (2023)

The problem with the Malthusian model

The Malthusian population model predicts unlimited population growth or inevitable extinction. The difference is based on whether the growth rateris it positive or negative.

No case is normally seen in genuine biological communities. Instead, what is often observed is that small populations often (though not always) increase in number, while very large populations tend to decrease in number. In both cases, a steady state is often reached after which no significant population change is observed unless significant environmental change is found. Therefore, a good population model should reproduce this behavior.

From a biological point of view, the missing feature in the Malthusian model is the idea of ​​carrying capacity. As a population increases in size, the ability of the environment to support the population decreases. As the population increases, per capita food availability decreases, waste can accumulate, and birth rates tend to decrease while death rates tend to increase. It seems reasonable to consider a mathematical model that explicitly incorporates the idea of ​​carrying capacity.

The logistic population model

ologisticsmodel, a slight modification of the Malthus model, is exactly that model. Like the Malthusian model, the logistic model includes a growth rater🇧🇷 This parameter represents the rate at which the population would grow if it were not burdened by environmental degradation.

A second parameter,k, represents the load capacity of the system under study. Carrying capacity is the population level at which the birth and death rates of a species precisely match, resulting in a stable population over time.

for rentXI)represents the population at the beginning of the time periodUEthe logistic model is:

X(i+1) - X(i) = r*X(i)*(1 - X(i)/K)

Where*represents multiplication.


  1. Determine Equilibrium PopulationsX(*)for this model They seem reasonable given the interpretations of the parameters.rmik?
  2. Assuming normal biological conditions, which equilibrium do you suspect is stable? unstable?
  3. Using the Stella version of the Malthus model as a starting point, create a Stella model for the logistic model of population growth.
  4. Defining parameter valuesr=0,07miK = 1000, and using an initial population of 150, run your Stella logistic model for 200 years, plotting population against time. Observe the behavior of the population. Does it match your expectations?
  5. Try other values ​​ofr,kand initial populations. In particular, try with initial populations that are larger thankit's less thank🇧🇷 Try to identify when the model behaves correctly and when the model seems "broken".

bidirectional flows

In question 5 above, you may have noticed that the model works well when the initial population is below carrying capacity, but it seems to work for larger initial populations. In the model, these larger populations do not shrink toward carrying capacity, as might be expected. On the other hand, large populations do not change anything.

In fact, our expectations of the model are correct. The large initial populationshe mustdecrease towards carrying capacity. What is wrong is not our model or our understanding of the model. Rather, the difficulty lies in a major subtlety with the Stella software.

The subtlety is this: unless Stella is explicitly told otherwise, Stella forces all flows to be nonnegative. If the formula for a flow results in a negative number, Stella will simply set the flow to zero and continue. Most of the time this is not a problem, but as we saw above, it can sometimes present difficulties. If a model isn't behaving as expected, it's always a good idea to see if Stella's behavior could be a source of difficulty.

There are two ways to overcome this difficulty. The first is to inform Stella that a certain flow is bidirectional. In this case, if a formula for a flow produces a negative value, Stella will move the material in the opposite direction of its normal flow. In the case of our logistics model, the material will flow from the stock (warehouse) back to the cloud.

To do this, double-click the flow, just as you did when writing the formula for the flow. At the top left of the window there will be a line indicating whether the flow should be bidirectional or not. Clicking on the appropriate circle will allow the flow to move in one or two directions. After clicking the circle, click the OK icon to save the change.

Birth and death rates

The second and preferred way to deal with the one-way difficulty is to redesign the model. Instead of simply calculating the net change in the population (which can be positive or negative), rearrange the model to track birth and death rates, both of which are probably not negative.

This revamp of the model involves a small change to the Stellamodel we previously developed. The new model will still have a stock (reservoir) that contains the population level. However, there will be two flows, one associated with births and the other associated with deaths. To create a flow outside of the stock, select the flow icon on the modeling level, place the cursor inside the stock, and hold down the mouse button while dragging the cursor outside of the stock. When you release the mouse button, a flow will be established from stock to a cloud. This stream should be tagged asdeceasedwhile the flow to stock should be tagged again as something likebirths.

odeceasedthe flow has no formula in it. The formula in the original flow, thebirthsflow, it needs to be changed. What formulas should go into these flows? The answer comes from rearranging the logistic equation into its positive and negative parts. We demonstrate this procedure here:

X(i+1) - X(i) = r*X(i)*(1 - X(i)/K)
X(i+1) - X(i) = r*X(i) - r*X(i)*X(i)/K

The first term of this formula,r*X(yo), can be interpreted as the birth rate in this model, while the second term,r*X(i)*X(i)/Kcan be interpreted as the mortality rate. Taking into account that the birth rate depends only onrwhile the mortality rate depends on bothrmik, the appropriate connectors can be created and deleted to place the required formula in each sequence.

This approach of explicitly determining birth and death rates in a population model is routinely implemented because it gives the modeler the opportunity to examine the model components to see if the birth and death formulas are reasonable. It also gives experimenters a way to break the larger problem of population dynamics into two simpler parts. separate death process.

The analysis that can be done by separating the processes of birth and death is well illustrated by the logistic model. The birth rate seems quite reasonable. It only depends on the birth rate per capitar🇧🇷 It may seem strange to some that the per capita birth rate does not decrease as the population increases. This is a weakness of the logistic model, and various attempts to correct this problem have resulted in many variations on the logistic model.

A more serious weakness in the logistic model becomes apparent when the death rate formula is considered. The number of deaths, given by


increases as the population increases. This represents the effects of congestion. load capacitykplays an important role in this formula, as it should. However, it seems strange thatr, the per capita birth rate appears in this formula. It does not seem reasonable that the birth ratedirectlyaffect the number of deaths that occur in a population. Many take this to be an indication that the logistic model is probably not a good representation of biological reality.

Despite the weaknesses illustrated above, the logistic model is often used in biological modelling, either as the basis for a more complicated model or as a rough model when the details of a population's dynamics are not known.


  1. Change the flow in the original Stella logistic model to be bidirectional, and repeat the previous exercises with initial populations above carrying capacity. Does the model now live up to expectations?
  2. Repeat the above using the explicit birth-death rate model discussed in the article. Does this model live up to expectations?
  3. Compare the results of these two exercises. If the results differ, explain the difference. If the results are identical, explain why they should be identical.


What is logistic population model? ›

The logistic model. Verhulst proposed a model, called the logistic model, for population growth in 1838. It does not assume unlimited resources. In- stead, it assumes there is a carrying capacity K for the population. This carrying capacity is the stable population level.

How do you find the logistic model for population growth? ›

An important example of a model often used in biology or ecology to model population growth is called the logistic growth model. The general form of the logistic equation is P(t) = \frac{KP_0e^{rt}}{K+P_0(e^{rt}-1)}.

What is the best model for population growth? ›

Per capita rate of increase (r)

r=(birth rate+immigration rate)–(death rate and emigration rate). If r is positive (> zero), the population is increasing in size; this means that the birth and immigration rates are greater than death and emigration.

Which population growth model is most realistic? ›

Logistic growth describes a model for population growth that takes into account carrying capacity, and is therefore a more realistic model for population growth. According to the logistic growth model, a population first grows exponentially because there are few individuals and plentiful resources.

What is the purpose of a population model? ›

Population models are used to determine maximum harvest for agriculturists, to understand the dynamics of biological invasions, and for environmental conservation. Population models are also used to understand the spread of parasites, viruses, and disease.

What are the 4 types of logistics? ›

Logistics can be split into five types by field: procurement logistics, production logistics, sales logistics, recovery logistics, and recycling logistics.

How do you calculate population model? ›

Population Growth Calculation

To calculate the Population Growth (PG) we find the difference (subtract) between the initial population and the population at Time 1, then divide by the initial population and multiply by 100.

How do you calculate logistic model? ›

To estimate a logistic regression we need a binary response variable and one or more explanatory variables. We also need specify the level of the response variable we will count as success (i.e., the Choose level: dropdown). In the example data file titanic , success for the variable survived would be the level Yes .

Why is the logistic growth model more realistic? ›

The logistic growth model is more realistic because habitat has limited resources. So, the growth shows initially a lag phase, followed by phases of acceleration and deceleration and finally the population density reaches the carrying capacity.

Why is logistic growth unrealistic? ›

The Logistic Growth Model:

The population sequence grows without bound, increasing by the same percentage at each stage. Of course, this is unrealistic. Resources will eventually run out in any habitat, prohibiting such unlimited growth, so one would expect the population to eventually level off.

Which method of population estimation is the most accurate? ›

Counting all individuals in a population is the most accurate way to determine its size.

Why is the population growth model unrealistic? ›

In the resulting model the population grows exponentially. In reality this model is unrealistic because envi- ronments impose limitations to population growth. A more accurate model postulates that the relative growth rate P /P decreases when P approaches the carrying capacity K of the environment.

Is logistic or exponential more realistic? ›

Logistic growth model is more realistic than exponential growth model.

Is logistic or exponential growth more realistic? ›

The logistic model is one step in complexity above the exponential model. It is more realistic and is the basis for most complex models in population ecology. Don't forget, though, that even this model simplifies the true complexities found in population biology.

What are the types of population models? ›

Three major types of population models are presented: continuous-time models, discrete-time models and stochastic models.

What are the two population models? ›

Two major models used by population ecologists to measure population growth are the exponential growth model and the logistical growth model.

Is the logistic growth model an accurate predictor of human population growth? ›

The logistic growth model is more accurate than other models in determining population growth because of the effect of the carrying capacity. The carrying capacity is the concept that resources are always limited because the environment can only support a certain number of individuals in a population.

What are the 5 P's of logistics? ›


What are the 3 P's of logistics? ›

Supply Chain and Risk Management: “3Ps” – Predictive, Proactive, Prescriptive - Spend Matters.

What are the 3 main logistics objectives? ›

There are five main objectives of the logistics management process:
  • Minimize Manufacturing Costs.
  • Efficient Flow of Operations.
  • Better Communication Flow.
  • Provides Competitive Edge.
  • Better Inventory Management.
  • Logistics Management Solution.
25 Feb 2020

What questions can Logistic Regression answer? ›

There are 3 major questions that the logistic regression analysis answers – (1) causal analysis, (2) forecasting an outcome, (3) trend forecasting. The first category establishes a causal relationship between one or more independent variables and one binary dependent variable.

Which solver is best for Logistic Regression? ›

The SAGA solver is a variant of SAG that also supports the non-smooth penalty L1 option (i.e. L1 Regularization). This is therefore the solver of choice for sparse multinomial logistic regression. It also has a better theoretical convergence compared to SAG.

What math is used in logistics? ›

The maths of logistics starts with algebra – linear algebra, to be precise. This is algebra where the variables (data about warehouse stock, for example) tend to be processed in ways that don't depend on the square, the cube or any other power. So y = 4x would be an operation in linear algebra; y = 4x2 would not.

What are the 4 ways to determine population size? ›

Here we compare estimates produced by four different methods for estimating population size, i.e. aerial counts, hunter observations, pellet group counts and cohort analysis.

How is population value calculated? ›

The population mean can be calculated by the sum of all values in the given data/population divided by a total number of values in the given data/population.

How is population data calculated? ›

Population estimates are dependent on the demographic components of change: mortality, fertility, and migration. Estimates of mortality, fertility, and migration are derived from data available from censuses, surveys, registration systems, and other administrative records.

What is a good sample size for logistic regression? ›

In conclusion, for observational studies that involve logistic regression in the analysis, this study recommends a minimum sample size of 500 to derive statistics that can represent the parameters in the targeted population.

What is one limitation of the logistic growth model? ›

Logistic Growth models tend toward being difficult to construct for a few reasons: Ambiguity in a modeling environment's carrying capacity is a big issue in logistic growth models. The carrying capacity of a given modeling environment may not be known, or may be highly variable.

How can you improve the accuracy of a logistic model? ›

Tips and Tricks to improve model precision
  1. Handling Null/Missing Values.
  2. Data Visualization.
  3. Feature Selection and Scaling.
  4. 3A. Feature Engineering.
  5. 3B. Feature Transformation.
  6. Use of Ensemble and Boosting Algorithms.
  7. Hyperparameter Tuning.
14 Jun 2021

How do you know if a logistics model is good? ›

It examines whether the observed proportions of events are similar to the predicted probabilities of occurence in subgroups of the data set using a pearson chi square test. Small values with large p-values indicate a good fit to the data while large values with p-values below 0.05 indicate a poor fit.

Why is logistic model better than exponential? ›

The main difference between exponential and logistic growth is that exponential growth occurs when the resources are plentiful whereas logistic growth occurs when the resources are limited. The exponential growth is proportional to the size of the population. It is influenced by the rate of birth and the rate of death.

Which growth curve is more realistic? ›

So, the correct answer to the question is 'Sigmoid'.

Does logistic growth have a limit? ›

Logistic growth

The logistic function models the exponential growth of a population, but also considers factors like the carrying capacity of land: A certain region simply won't support unlimited growth because as one population grows, its resources diminish. So a logistic function puts a limit on growth.

Which of the three types of estimating methods is most accurate? ›

Bottom-Up Estimate

In fact, bottom-up estimates are the most accurate type of cost estimate.

What is the best estimator for population proportion? ›

  • The best estimate of the population mean is the sample mean: ...
  • The best estimate for the population's mean SBP is the sample mean (that is, 140 mm Hg). ...
  • Again the best estimation of the population's proportion is the samples proportion.

What are the two methods used to estimate population size? ›

A population estimate is a calculation of the size of a population for a year between census periods or for the current year. There are two types of estimation techniques: inter-census and post-census.

What is the simplest model of population growth? ›

Aside the basic linear models for either growth or decay of a set of independent individuals, the simplest model for population dynamics is the logistic interaction among its members.

What are the 4 main challenges of population growth? ›

Without taking action now, billions of people across the world will face thirst, hunger, slum conditions and conflict in response to droughts, food shortages, urban squalor, migration and ever depleting natural resources, while capacity tries to catch up with demand.

What are the three challenges of population growth? ›

Populations change in response to three driving factors: fertility – how many people are born; mortality – how many people die; and migration – how many people leave or enter the population.

Does human follow exponentially or logistically? ›

The world's human population is currently experiencing exponential growth even though human reproduction is far below its biotic potential ([link]). To reach its biotic potential, all females would have to become pregnant every nine months or so during their reproductive years.

Is it realistic that most populations will exhibit exponential growth forever? ›

Realistically, no population grows in an exponential manner forever. Populations are limited by food, space, light, waste build-up, and by populations of other organisms. Once a particular resource becomes limiting, population growth will slow and eventually stop.

Is there growth faster than exponential? ›

Factorials grow faster than exponential functions, but much more slowly than doubly exponential functions. However, tetration and the Ackermann function grow faster.

Which growth model is considered realistic one? ›

Assertion: The logistic growth model is considered to be more realistic.

Why is logistic growth a more realistic representation of population growth than exponential growth? ›

Logistic growth is a more accurate depiction of population growth because habitats contain limited resources which prevent populations from continually growing exponentially once the population hits it's carrying capacity.

What does a logistic model represent? ›

In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables.

What is the logistic growth model used for? ›

The logistic growth model is more accurate than other models in determining population growth because of the effect of the carrying capacity. The carrying capacity is the concept that resources are always limited because the environment can only support a certain number of individuals in a population.

What are logistics models? ›

The term “logistics model” might sound complicated at first. It's nothing other than all the actions you undertake to stock and sell products in your store. There are 3 main logistics models: warehouse model, fulfilment and dropshipping. Every single one of them has its pros and cons.

What is logistic modeling in biology? ›

The logistic model reveals that the growth rate of the population is determined by its biotic potential and the size of the population as modified by the natural resistance, or, in other words, by all the various effects of inherent characteristics, that are density dependence Pearl and Reed, 1920.

What is logistic example? ›

Logistics refers to what happens within one company, including the purchase and delivery of raw materials, packaging, shipment, and transportation of goods to distributors, for example.

Why is the logistic population growth curve? ›

When resources are limited, populations exhibit logistic growth. In logistic growth, population expansion decreases as resources become scarce, leveling off when the carrying capacity of the environment is reached, resulting in an S-shaped curve.

Why logistic strategy is important? ›

Logistics Plans Help You to Deliver Products on Time

They provide tracking information so customers can prepare for the arrival of their orders and this feature helps them measure their expectations. By working with a third-party logistics provider, you can stay competitive and meet your customer expectations.

What are the 3 types of population growth? ›

And while every population pyramid is unique, most can be categorized into three prototypical shapes: expansive (young and growing), constrictive (elderly and shrinking), and stationary (little or no population growth). Let's take a deeper dive into the trends these three shapes reveal about a population and its needs.

What is logistics simple answer? ›

What Are Logistics? Logistics refers to the overall process of managing how resources are acquired, stored, and transported to their final destination. Logistics management involves identifying prospective distributors and suppliers and determining their effectiveness and accessibility.

What logistic means? ›

Logistics is used more broadly to refer to the process of coordinating and moving resources – people, materials, inventory, and equipment – from one location to storage at the desired destination. The term logistics originated in the military, referring to the movement of equipment and supplies to troops in the field.

Why is it called logistic function? ›

Logistic comes from the Greek logistikos (computational). In the 1700's, logarithmic and logistic were synonymous. Since computation is needed to predict the supplies an army requires, logistics has come to be also used for the movement and supply of troops".


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