# Is it possible to mathematically model the growth of plants?

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Are there mathematical equations that represent/model the growth of plants? For example, if I have a Bellis perennis and I want to know how it develops in a given environment (Sun is shining, defined amount of water etc). How can one say if it grows a certain height and leafs and all other organs? I don't want a generic solution for this like using L-System and stuff, I just want to have a source where I can research growth rules for a given plant, flower or tree.

Background:
I'm making a kind of mini-game for a computer-graphics course at university. Think of it like a 1st-person shooter where you take a seed and plant it into the earth. The plant starts to grow and you will see it grow like in real life. Of course there are some environmental constraints that should be taken into account:
A. Time
B. (Sun)Light
C. Water
D. Other nutrition stuff like fertilizer

Don't bother about the technical aspects, i just want to know about the biological aspects of plant-growth. Let's assume we have a sunflower plant. As it starts to grow I get a couple of questions:
1. How fast does the plant stem grow according to a well defined amount of luminosity and water?
2. When do the first leaves develop?
3. How much thickness the stem gains per day/hour/minute?
4. What happens when suddenly there is less amount of luminosity and/or water? How the growth-rate will be affected?

These are only a few questions that arise when i think about this topic. Therefore i want to have a mathematical description of plant-growth under the environmental constraints mentioned above. I don't need a generic approach, but it would be nice to see such a formalism for e.g. a sunflower or Bellis Perennis. I also don't care about things that won't be able to see in the scene, like micro-biological-level - i just care about the whole architecture of the plant (stem, branches, leafes, fruits, (root)).

I did a lot of search on this topic including the computer modeling side: http://algorithmicbotany.org/papers/
https://www.informatik.uni-konstanz.de/en/deussen/publications/
But these sources focus more on techniques on how to generate images instead of the biological aspects.
Another book I've searched too solve my questions is "On Growth and Form" by Darcy Wentworth Thompson, but thats too general for my needs.

So are there any formal descriptions of plant-growth that take luminosity, water etc into account?

## A new mechanism behind continuous stem cell activity in plants

Figure 1: Diagrams of the vascular genetic expression network (Purple= Xylem cells, Green= Phloem Cells, Blue= Vascular stem cells). A. Vascular development during the plant’s secondary growth. B. The vascular cell induction culture system ‘VISUAL’. C. The constructed vascular gene expression network. Each dot indicates a gene and the lines show strong mutual relationships between them. Credit: Kobe University

An inter-university research group has succeeded in constructing the gene expression network behind the vascular development process in plants. They achieved this by performing bioinformatics analysis using the 'VISUAL' tissue culture platform, which generates vascular stem cells from leaf cells. In this network, they also discovered a new BES/BZR transcription factor, BEH3, which regulates vascular stem cells. In addition, they illuminated a novel vascular cell maintenance system whereby BEH3 competes with other transcription factors from the same BES/BZR family in order to stabilize vascular stem cell multiplication and differentiation.

The joint research group consisted of scientific researcher Furuya Tomoyuki and Associate Professor Kondo Yuki et al. (of Kobe University's Graduate School of Science), Kyushu University's Professor Satake Akiko, Specially Appointed Professor Tanokura Masaru and Specially Appointed Associate Professor Miyakawa Takuya (of the University of Tokyo's Graduate School of Agricultural and Life Sciences), and Associate Professor Yamori Wataru (of the University of Tokyo's Institute for Sustainable Agro-ecosystem Services).

The researchers hope to identify more regulatory factors for stem cells, which will contribute towards our understanding of the molecular basis behind continuous stem cell activity in plants.

These research results were published in the American plant sciences journal The Plant Cell on June 1, 2021.

• The researchers extracted a total of 394 genes specific to vascular stem cells from extensive gene expression datasets. Among these, they discovered BEH3, a novel stem cell regulatory factor belonging to the BES/BZR family of transcription factors.
• They discovered that unlike the other BES/BZR transcription factors, BEH3 has almost no functional domains and competitively inhibits the activity of these other factors.
• The research group showed that this competitive relationship between BES/BZR transcription factors stabilizes the multiplication and differentiation of vascular stem cells, illuminating the regulatory system that maintains vascular stem cells' continuous activity.
Figure 2: Model diagram showing the competitive relationship between BEH3 and other BES/BZR transcription factors. BEH3 and the other BES/BZR transcription factors (here represented by BES1) compete with each other to bind to the DNA motif BRRE, regulating downstream gene expression. Credit: Kobe University

Plants take form by self-replicating their stem cells and differentiating these stem cells so that they have specialized functions for constructing parts of the plant, such as its organs and tissues. Unlike animals, plants continue to regenerate and grow by producing stem cells throughout their life. For example, trees such as cryptomeria can have long lifespans (the Jomon Cedar Tree on Japan's Yakushima Island is at least 2000 years old), and each year they promote secondary growth which results in another tree ring around their trunks. This secondary growth is occurs inside a region of meristem tissue called the cambium layer where vascular stem cells multiply and differentiate into xylem cells and phloem cells, enabling the trunk to grow wider. In other words, plant must continuously produce vascular stem cells throughout their lives in order to keep growing, and it is vital for them to maintain the balance between stem cell multiplication and differentiation.

In recent years, studies using the model plant Arabidopsis thaliana have been conducted into how the multiplication and differentiation of stem cells are regulated from the perspectives of genetics, life sciences and informatics research. However, the mechanism by which plants regulate and maintain the appropriate balance of stem cells has yet to be understood.

Research Methodology and Findings

In order to analyze the process by which vascular stem cells differentiate into xylem cells and phloem cells (Figure 1), Associate Professor Kondo et al."s research group developed the tissue culture system 'VISUAL' to artificially generate stem cells from leaf cells. VISUAL has many benefits that make it suitable for research on vascular stem cells, for example, it is easy to genetically analyze plants that have a particular gene function removed (i.e. mutants) and it is also possible to observe the temporal progression of vascular stem cell differentiation. In this study, the researchers collected data on multiple mutants and carried out large-scale analyses of gene expression at various time points. They conducted gene co-expression network analysis on similarities in the expression patterns to evaluate the relationship between different genes. From this analysis, they succeeded in identifying the distinctive groups of genes in xylem cells, phloem cells and vascular stem cells (Figure 1). Using VISUAL, this research group previously revealed that the BES/BZR transcription factors BES1 and BZR1 play an important role in vascular stem cell differentiation. This time, they identified another BES/BZR transcription factor, BEH3, in the vascular stem cell gene group through network analysis, and also examined its vascular stem cell suppressing function.

Figure 3: A model of vascular stem cell regulation based on this research. Competition within the BES/BZR transcription factor family robustly regulates the balance between vascular stem cell multiplication and differentiation, contributing towards the maintenance of continuous stem cell activity. Credit: Kobe University

Next, the researchers investigated vascular formation using mutants with BEH3's function removed. They found that the mutants had large variations in vascular size compared to the wild type (non-mutant plant) and concluded that BEH3 stabilizes vascular stem cells. The research group had previously found that strengthening the function of BES1 (which promotes vascular cell differentiation) caused the number of vascular cells to decrease, however they found that when they strengthened the function of BEH3 opposite occurred and the number of vascular stem cells increased. Upon researching this further, the research group discovered that even though BEH3 could bind to the same DNA motif as the other BES/BZR transcription factors, BEH3's ability to regulate the expression of downstream genes was significantly weaker. These results showed that BEH3 hinders the activity of other BES/BZR transcription factors (Figure 2), and the researchers inferred from this relationship that BEH3's function in vascular stem cells is opposed to that of the factors in the same family, including BES1. A mathematical model was used to verify and simulate this competitive relationship between BEH3 and the other BES/BZR transcription factors, and the results indicated that the presence of BEH3 in vascular stem cells contributes towards stabilizing vascular size (Figure 3).

There are thought to be many important gene candidates in this research group's vascular stem cell gene expression network that will contribute towards understanding of vascular development and functions. It is hoped that the valuable information obtained through this study will accelerate vascular research. In addition, further illuminating the relationships between BEH3 and other BES/BZR transcription factors and their respective differences will deepen our understanding of the mechanism by which plants maintain the balance between stem cell multiplication and differentiation.

In the future, this knowledge could contribute towards biomass production techniques, and other areas that require large-scale stable plant growth.

## Examples of the Null Hypothesis

A researcher may postulate a hypothesis:

It is important to carefully select the wording of the null, and ensure that it is as specific as possible. For example, the researcher might postulate a null hypothesis:

There is a major flaw with this H0. If the plants actually grow more slowly in compost than in soil, an impasse is reached. H1 is not supported, but neither is H0, because there is a difference in growth rates.

If the null is rejected, with no alternative, the experiment may be invalid. This is the reason why science uses a battery of deductive and inductive processes to ensure that there are no flaws in the hypotheses.

Many scientists neglect the null, assuming that it is merely the opposite of the alternative, but it is good practice to spend a little time creating a sound hypothesis. It is not possible to change any hypothesis retrospectively, including H0.

## History

The Gompertz [1] model has been in use as a growth model even longer than its better known relative, the logistic model [2]. The model, referred to at the time as the Gompertz theoretical law of mortality, was first suggested and first applied by Mr. Benjamin Gompertz in 1825 [1]. He fitted it to the relationship between increasing death rate and age, what he referred to as “the average exhaustions of a man’s power to avoid death”, or the “portion of his remaining power to oppose destruction”. The insurance industry quickly started to use his method of projecting death risk. However, Gompertz only presented the probability density function.

It was Makeham [15] who first stated this model in its well-known cumulative form, and thus it became known as the Gompertz-Makeham (or sometimes Makeham-Gompertz) model, a name we encounter for the first time in Greenwood’s [16] discussions. The first attempt to use a least-squares method for the Gompertz model to find the best curve, was attempted, e.g. [17] [18]. However, they did not linearize the model, as is done later, but only log-transformed the values (dependent variable) to make it easier to determine the sum of squares. This method seems to have been used until the 1940s [19], when Hartley [20] proposed to and first explained how to linearize the Gompertz model.

From the 1920s the cumulative Gompertz-Makeham model also rapidly became a favourite in fields other than that of human mortality, for example in forecasting the increase in demand for goods and services, sales of tobacco, growth in railway traffic, and the demand for automobiles [21][22]. Wright [23] was the first to propose the Gompertz model for biological growth, and the first to apply it to biological data was probably Davidson [24] in his study of body-mass growth in cattle. In 1931 Weymoth, McMillin, and Rich [25] reported the Gompertz model to successfully describe the shell-size growth in razor clams, Siliqua patula, and Weymouth and Thompson [26] reported the same for the Pacific cockle, Cardium corbis. Soon, researchers began to fit the model to their data by regression, and over the years, the common [15] Gompertz model became a favourite regression model for many types of growth of organisms, such as dinosaurs, e.g. [27] [28], birds, e.g. [13] [29] [30] [31], and mammals e.g. [32] [33] including those of marsupials, e.g. [34] [35]. The Gompertz model is also frequently applied to model growth in number or density of microbes [36, 37], growth of tumours [4, 38, 39], and the survival of cancer patients [40].

Several different re-parameterisations of the traditional cumulative Gompertz model are in use. One of the more important was suggested by Zwietering and colleagues [6] for modelling growth in number of bacteria, and is currently one of the most common models in microbial growth [7, 41,42]. Another prominent re-parametrisation of the Gompertz model is the Gompertz-Laird model, proposed by Laird and fitted to tumor growth data [4]. This model is considered especially useful when we want to discuss the initial value (starting point on the x-axis), and it is greatly used also for describing growth in birds and animals, especially poultry [e.g. 9, 43, 44, 45], and livestock [e.g. 46]. However, the model parameters are not easily interpretable without being converted to more useful measurements.

In addition to ordinary monotonically increasing Gompertz re-parameterisations, modellers of microorganisms in food have developed a number of modified monotonically decreasing Gompertz models for (thermal, pressure, or electric field) inactivation kinetics. We will not discuss any of these here, as their interest is limited to this particular type of “growth” studies.

## New Challenges in Systems Biology: Understanding the Holobiont

More than a century ago, C. Darwin formulated the Theory of Evolution based on Natural Selection without knowing how information is transmitted from one generation to another. In Darwin's theory, the concept of individual is of fundamental importance: Individuals mutate, and advantageous mutations that improve adaptation to the environment are passed on to the next generations, perpetuating an improvement cycle. Now we know that mutations occur in the genomes of single individuals, and those mutations, if beneficial (or neutral), can be passed on to the next generations. However, in the Theory of Evolution formulated by Darwin and accepted for more than a century, the concept of individual is taken for granted: an individual is a single entity that contains (in its genome) all the necessary information to generate organisms with similar (or identical) phenotypes. However, the very concept of “individual,” on which Darwin's theory relies, has recently been challenged (Bordenstein and Theis, 2015 Suárez and Stenzel, 2020).

Today, we know that every multicellular organism is an ecological system. When such organisms emerged on Earth several millions of years ago, the planet was already crowded with microbial life. Consequently, the emergence and evolution of plants and animals were not only carried out in the presence of microbes but, in many cases, such evolution was only possible because of these microorganisms (Zilber-Rosenberg and Rosenberg, 2008 Alegado et al., 2012 Carpenter, 2012 Gaulke et al., 2016). Recent studies have unequivocally shown that there is a great variety of microbes living in plant and animal hosts, the totality of which is known as the host's microbiota. For humans, it has been estimated that the number of bacterial cells inhabiting a human body is comparable with the number of body cells (Abbot, 2016 Sender et al., 2016), and that the microbiota strongly interacts with its host, regulating important metabolic functions at all levels, including the genetic level (Bates et al., 2006 Wang et al., 2018).

The human microbiota helps in the development of the immune system, the fortification of bones, the digestion of food, the regeneration of skin, and affects many other essential metabolic functions. Furthermore, there is a strong correlation between the microbiota's composition and the occurrence of complex diseases such as obesity, diabetes, cancer, metabolic syndrome, inflammatory bowel disease, allergies to gluten and lactose, and even cognitive and neurological disorders such as autism and schizophrenia (Wang et al., 2017, 2018 Fan and Pedersen, 2021). Multicellular organisms have coevolved with microbes and strongly depend on them (even bacteria have their microbiota, which consists of viruses). Therefore, the question arises: what is an individual?

To emphasize that every individual is, in fact, a complex ecosystem, Rosenberg and his collaborators (Theis et al., 2016 Roughgarden et al., 2018) have proposed to use the term holobiont [first introduced by Lynn Margulis in a different context (Guerrero et al., 2013)] to refer to the host organism together with its microbiota. The holobiont is more than just an aggregate of cells that live together, as all cells in an organism carry the same genetic information in all organs. By contrast, in a holobiont, cells with different genetic compositions (even from different kingdoms), live together, interact, and exhibit complex dynamical behaviors. Imbalances in the holobiont ecosystem may lead to dysbiosis, a medical condition that may cause unwanted symptoms and, as mentioned above, has been associated with the onset of serious diseases. For a long time, predisposition to such diseases was attributed only to particularities in the genetic material of the host organism. However, if the microbiota's composition is strongly correlated with a complex disease, say obesity, it is possible that the microbiota that favors the occurrence of that disease is being transmitted from mother to offspring (Veigl et al., 2019). If that were the case, then the inheritance of phenotypes could be attributed not only to the genetic composition of the host organism but also to the transmission of microbes across generations.

Such connections would potentially challenge one of the central dogmas in the current Theory of Evolution, making Lamark's type of inheritance possible (Rosenberg et al., 2009). Indeed, there is strong evidence suggesting that. For instance, we know that obesity, diabetes, or breast cancer are inherited with high probability, as shown by familial studies. However, genetic markers for these diseases are yet to be discovered (a problem known as “the missing heritability”) (Manolio et al., 2009 Eichler et al., 2010 Génin, 2019). One possible explanation may hide in plain sight: our efforts to answer these questions are focused on looking for genetic markers only in the human genome (namely just in a tree of the entire forest), while we now know that the human microbiota is strongly correlated with the emergence of complex diseases. Therefore, when looking for genetic signatures of these diseases we should be looking at the entire forest that is, consider not only the human genome but also the totality of genomes of all organisms in the microbiota—the microbiome (Sandoval-Motta et al., 2017).

Treatments for some illnesses, like the irritable bowel syndrome, that aim to rebalance the microbiota by transplanting fecal matter from a healthy person to a sick person (therapy known as �l transplant” or �teriotherapy”) date back to ancient China (De Groot et al., 2017). More recent studies (Woodworth et al., 2019) have shown that this approach is effective in reducing intestinal colonization with antibiotic-resistant bacteria, even though we may not yet fully understand the mechanism at the molecular or genetic levels. This suggests that transplanting bacteria from healthy to sick people may also work for other complex diseases such as obesity, cancer, or autism. The answer is yet to be discovered.

The strong symbiotic interactions between bacterial communities and their hosts (including humans) within the holobiont, were discovered only recently with the development of high-throughput sequencing techniques (Visscher et al., 2017). Most bacteria in the microbiota cannot be cultivated outside their host organism—let by themselves, these bacterial communities establish interactions dominated by competition, resulting in the dominance of one bacterial strain. It is the host that regulates these otherwise competing interactions, allowing different bacterial communities to inhabit the same organism (Foster and Bell, 2012 Coyte et al., 2015). As already mentioned above, bacteria in turn help the host in carrying out many different metabolic functions. High-throughput sequencing techniques provide a way to reveal the structure of complex ecosystems hidden inside the host and unveil those symbiotic interactions (Wang and Jia, 2016). The host, in turn, influences the composition of the microbiota. In the case of humans, the type of food we eat exerts an influence. A concrete example can be found in some bacteria living in the gut which transform carbohydrates into serotonin, a neurotransmitter associated with happiness (Stasi et al., 2019). Therefore, eating carbohydrates makes us happy, which creates a positive feed-forward cycle between us and gut bacteria that can lead to obesity and diabetes (It is still a matter of debate how the serotonin produced in the gut can cross the blood-brain barrier and reach the neurons in the brain).

It has been almost a decade since the pioneering work by Turnbaugh and his collaborators, who demonstrated that the bacteria in the human gut can determine important phenotypic traits (obesity) in mice (Turnbaugh et al., 2006). Since then, we have learned a lot about the symbiotic relationships between microbes and multicellular organisms. Mathematical models have also been deployed to investigate holobiont selection as an evolutionary force (Huitzil et al., 2018 Roughgarden, 2019). But we have only scratched the surface. There are many problems and questions that remain unsolved. In our opinion, one of the greatest challenges that we face for the twenty-first century regarding Systems Biology is to develop mathematical and computational models that help us understand the holobiont as a complex ecosystem. How and why such strong symbiotic relationships between bacteria and multicellular organisms have appeared throughout evolution? Did they appear just because it was possible? If not, what evolutionary advantages emerge from these symbiotic interactions? Will the concept of “individual” in evolutionary theories need to be reformulated (or replaced) to take into account the holobiont as a complex ecosystem subject to selection? What complex diseases could be cured by altering the microbiota's composition of the patient? Are there phenotypes that can be “inherited” through the microbiota and not through parental DNA? Is it possible to engineer a “healthy microbiota” or a microbiota that favors the emergence of desired phenotypes in the host organism?

The list of major challenges and unanswered questions is already long. And as we unravel some of these questions in the future, the list will get even longer. The Systems Biology community around the world faces a significant challenge—we need experimental methods, mathematical models, and computational approaches that combine the best available data from genomics, metabolomics, and proteomics with existing knowledge in the life sciences to help us understand the evolution, dynamics, and behavior of holobionts not as individuals, but as complex ecosystems.

## Nutrient Levels

To grow plants hydroponically, it is helpful to maintain proper nutrient levels. For small scale systems, pre-mixed plant food is an easy way to add additional nutrients (available at a local plant nursery or hardware store). Generally, a nutrient mix contains a balance of the three main elements important for plant growth: nitrogen (N), phosphorus (P), and potassium (K). Other nutrients may include calcium (Ca), magnesium (Mg), sulphur (S), iron (Fe), manganese (Mn), copper (Cu), zinc (Zn), molydenum (Mo), boron (B), and chlorine (Cl).

## Population Growth & Regulation: Geometric, Logistic, Exponential

Humans have a large impact on the global environment: Our population has grown explosively, along with our use of energy and resources.

Human population reached 6.8 billion in 2010, more than double the number of people in 1960.

Our use of energy and resources has grown even more rapidly.

From 1860 to 1991, human population quadrupled in size, and energy consumption increased 93-fold.

Predicted Decrease in Growth of Population

Ecological footprint: Total area of productive ecosystems required to support a population. Uses data on agricultural productivity, production of goods, resource use, population size, and pollution. The area required to support these activities is then estimated.

****Population: 6.6 billion, a 40% overshoot of carrying capacity.

One of the ecological maxims is: “No population can increase in size forever.”

The limits imposed by a finite planet restrict a feature of all species: A capacity for rapid population growth.

Ecologists try to understand the factors that limit or promote population growth

A life table is a summary of how survival and reproductive rates vary with age.

Information about births and deaths is essential to predict future population size.

Sx = survival rate: Chance that an individual of age x will survive to age x + 1.

lx = survivorship: Proportion of individuals that survive from birth to age x.

Fx = fecundity: Average number of offspring a female will have at age x.

Birth and death rates can vary greatly between individuals of different ages.

In some species, age is not important, e.g., in many plants, reproduction is more dependent on size (related to growth conditions) than age.

Life tables can also be based on size or life cycle stage.

Survivorship curve: Plot of the number of individuals from a hypothetical cohort that will survive to reach different ages.

Survivorship curves can be classified into three general types.

Type I: Most individuals survive to old age (Dall sheep, humans).

Type II: The chance of surviving remains constant throughout the lifetime (some birds).

Type III: High death rates for young, those that reach adulthood survive well (species that produce a lot of offspring).

A population can be characterized by its age structure—the proportion of the population in each age class.

Age structure influences how fast a population will grow.

If there are many people of reproductive age (15 to 30), it will grow rapidly.

## Human Population Growth

Global human population growth is around 75 million annually, or 1.1% per year. The global population has grown from 1 billion in 1800 to 7 billion in 2012. It is expected to keep growing, though predictions differ as to when and if this growth will plateau.

Figure (PageIndex<1>): The estimated growth of the human population from 10,000 BCE&ndash2000 CE.: The human population has grown most sharply in the past 200 years.

The &ldquopopulation growth rate&rdquo is the rate at which the number of individuals in a population increases in a given time period as a fraction of the initial population. Specifically, population growth rate refers to the change in population over a time period, often expressed as a percentage of the number of individuals in the population at the beginning of that period. This can be written as the formula:

population growth rate=P(t2)&minusP(t1)P(t1)population growth rate=P(t2)&minusP(t1)P(t1)

Globally, the growth rate of the human population has been declining since 1962 and 1963, when it was 2.20% per annum. In 2009, the estimated annual growth rate was 1.1%. The CIA World Factbook gives the world annual birthrate, mortality rate, and growth rate as 1.89%, 0.79%, and 1.096% respectively. The last 100 years have seen a rapid increase in population due to medical advances and massive increase in agricultural productivity.

Each region of the globe has seen reductions in growth rate in recent decades, though growth rates remain above 2% in some countries of the Middle East and Sub-Saharan Africa, and also in South Asia, Southeast Asia, and Latin America. This does not mean that the population is declining
rather, it means the population is growing more slowly. However, some countries do experience negative population growth, mainly due to low fertility rates, high death rates and emigration.

According to the UN&rsquos 2010 revision to its population projections, world population will peak at 10.1 billion in 2100 compared to 7 billion in 2011. However, some experts dispute the UN&rsquos forecast and have argued that birthrates will fall below replacement rates (the number of births needed to maintain a stable population) in the 2020s. According to these forecasters, population growth will be only sustained until the 2040s by rising longevity, but will peak below 9 billion by 2050, followed by a long decline.

Figure (PageIndex<1>): Growing Population Rate and Resource Scarcity: Greater Los Angeles lies on a coastal Mediterranean Savannah with a small watershed that is able to support at most one million people on its own water as of 2015, the area has a population of over 18 million. Researchers predict that similar cases of resource scarcity will grow more common as the world population increases.

## 3. The Verhulst Model

The Malthus model was not good enough to describe the yeast population for a longer time because it does not consider the scarcity of resources. It is too simple and therefore one needs to add to it little more information to obtain better results.

As the Malthus model behaves well in the early stage of growth, one still relies on this model, but making a small change in it. Rewriting it with the introduction of a correction term in the ODE (6) yields to

This "term" should be zero (or near zero) when N is sufficiently small (reaching the Malthus model in this regime) and should be maximum when the population reaches a certain level. Verhulst [ 24 [24] P. Verhulst, Nouveaux Memoires de l’Academie Royale des Sciences et Belles Lettres de Bruxelles 18, 1 (1845). , 25 [25] P. Verhulst, Nouveaux memoires de l’Academie Royale des Sciences et Belles Lettres de Bruxelles, 20, 1 (1847). ] considered this corrective term to be proportional to N 2 ∕ K , where K is the carrying capacity of the population, which is the maximum size of the population that can be supported by the environment. Thus, with corrective term the Malthus ODE takes the form of the Verhulst model

When the population size approaches the carrying capacity ( N → K ), then d N ∕ d t = 0, i.e. the population stops growing. This model predicts, when t → ∞ , that the population saturates instead of blowing up, as in the Malthus model.

The solution of this model is obtained by integrating both sides of Eq. (9) (see appendix (A) for details), resulting in

Note in the fit of Fig. (1) that the Verhulst model is appropriate to describe the yeast population growth. The solution (10) describes the population dynamics of yeast fairly well, both for early and late time growth. One can say that the Verhulst model (although its simplicity, using only two parameters: the growth rate and carrying capacity) captures the essence of the yeast population growth. This good description occurs even disregarding most of the details involved in the dynamics. Nevertheless supplying more details, more parameters are need to describe the situation and perhaps impedes the theoretical treatment, hardening the understanding of the phenomenon. With this simple version, considering only phenomenological parameters, it is possible to describe this population quantitatively, with an analytic solution. In fact, the model explains quantitatively that the yeast population is growing rapidly in the beginning and then saturates due to the scarcity of resources.

## Forest Succession in More Detail

Forest succession is considered a secondary succession in most field biology and forest ecology texts but also has its own particular vocabulary. The forest process follows a timeline of tree species replacement and in this order: from pioneer seedlings and saplings to transition forest to young growth forest to mature forest to old growth forest.

Foresters generally manage stands of trees that are developing as part of a secondary succession. The most important tree species in terms of economic value are a part of one of several serial stages below the climax. It is, therefore, important that a forester manage his forest by controlling the tendency of that community to move toward a climax species forest. As presented in the forestry text, Principles of Silviculture, Second Edition, "foresters use silvicultural practices to maintain the stands in the seral stage that meets society's objectives most closely."