Laboratory work №1.

"Population dynamics".

Simulation of Population Dynamics Using the Calculation Program

Goal of the work: To study models of population dynamics with the help of a calculation program.

Approved for work

I've done the work

Work defended

2010 G.

1 THEORETICAL INTRODUCTION

According to the definition of the famous Russian ecologist S.S. Schwartz, population- this is an elementary grouping of organisms of a certain species, which has all the necessary conditions for maintaining its population for a long time in constantly changing environmental conditions.

Populations, like any biological open system, are characterized by a certain structure, growth, development, resistance to abiotic and biotic factors.

The most important indicator of the well-being of a population (resilience), its role in the functioning of a natural ecosystem is its abundance.

Population size is determined mainly by two phenomena - fertility and mortality, as well as migration.

fertility - the number of new individuals that appeared per unit of time as a result of reproduction. In the process of reproduction, the number of individuals increases; theoretically, it is capable of an unlimited increase in numbers.

There are various types of change in the number of individuals in a population depending on time (population dynamics). In the simplest cases, population dynamics can be described by simple mathematical models that allow predicting changes in the number of individuals.

1. Exponential population growth.

One of the earliest models of population growth was proposed by T. Malthus 1798, in wide famous work"On the Principles of Population". This model is called exponentialdependencies population growth (exponential growth curve). In this model, it is assumed unlimited amount of natural resources, available to individuals in the population and the absence of any limiting factors for population growth. Under such assumptions, the number of individuals in the population increases according to a power law, i.e. very fast and unlimited.

If denoted by n 0 the number of individuals in the population and the initial time (t 0 ), and through N t the number of individuals at some point in time t (t>t0). Then the change in the number ∆N over the time interval ∆ t. those. The population growth rate will be:

(1)

Expression (1) gives the average population growth rate. However, in population ecology, not the absolute average rate is used more often, but the growth rate per organism (specific rate):

(2)

This indicator allows you to compare the values ​​of the change in the number of populations of different sizes. In this case, the number is defined as the rate of increase by one individual in a certain time interval.

Passing to the limiting form of recording the velocity at
0 and
and introducing a new notation:

(3)

In expression (3) index r can be defined as the instantaneous specific population growth rate. For different populations of the same species, this indicator may have various meanings. The largest of all possible values ​​\u200b\u200b(r max) is called the biotic or reproductive potential of the population

Taking into account expression (3), the population growth rate can be described by the following expression

(4)

Differentiating expression (4), we obtain that at any time, under the condition r= withonst (growth rate constant) the number of individuals in the population will be equal to:
(5)

Formula (5) describes an exponential population growth model, which graphically has the shape of a curve (Fig. 1). The exponential growth model meets the conditions unlimited growth the number of individuals in the population.

Rice. 1. Exponential growth curve for the number of individuals in a population

1. Logistic growth model

The maximum population size that an ecosystem can sustain indefinitely under constant environmental conditions is called ecosystem capacity for this type.

Population change- this is the ratio between the biological potential (the addition of individuals) and the resistance of the environment (the death of individuals, mortality). Environmental resistance factors lead to an increase in mortality, and the abundance curve flattens out or even goes down if the population explosion has caused the depletion of vital ecosystem resources. The population growth curve with environmental resistance acquires S-figurative view (Fig. 2).

Rice. 2 . S-shaped population growth model

Thus, in natural conditions, unlimited growth is impossible and sooner or later the population will reach its limit, which is determined medium capacity(spatial, food, etc.). If we denote by the maximum possible number of individuals in a population a certain value K (medium capacity) and enter a correction factor that takes into account "resistance" environment growth in the form of a ratio:

,

then the equation for this case can be written as form:

(7)

The solution to this differential equation will look like

(8)

Where A - integration constant that determines the position of the function relative to the origin, it can be found from the expression (provided that r= const).

(9)

Expression (8) describes the so-called logistic growth curve(Fig. 2). This is the second simplest mathematical model of population dynamics under the condition of the upper limit of abundance and the resistance of the environment to population growth. According to this model population size at the firststage grows rapidly enough, but then the population growth rate slows down andbecomes infinitely small near the valueTO (the logistic curve asymptotically approaches the horizontal TO).

Subject: Winter tires. Region: Ukraine. Margin: 13%. Promotion period: 1.09 - 31.12 2012 vs 1.09 - 31.12 2013. spending: 42 389 UAH vs 131 341 UAH (including agency fees).

Although I am not a mathematician by education, I have sympathy for this science, so the article will use some, at first glance, complex mathematical terms.

The purpose of this article is to talk about one curious phenomenon: by doubling your advertising budget, you start earning not twice as much, but at 2.5, 3, etc. times more. Of course, up to a certain point. This phenomenon in mathematics is called exponential growth. An example of exponential growth would be the growth in the number of bacteria in a colony before the resource limit occurs.

For those of you who have dealt with compound interest, for example, when calculating income on deposits, it will immediately become clear what this is about, since compound interest is just another example of exponential growth. If you do not withdraw the accumulated funds from the deposit, then the income growth does not occur linearly, but exponentially. It’s the same with sales revenue growth: as the advertising budget increases, revenue grows exponentially. In this article, I would like to illustrate another phenomenon. It is because of this phenomenon that the contextual advertising department is no longer called that, but is called the paid traffic department. It's about about the synergy effect.

What is the synergy effect? Imagine an ideal situation: there is an online store, for its promotion in the first month only contextual advertising was used, which brought 20 sales, and in the second month only SEO promotion was used, which also gave 20 sales. In the third month, both contextual advertising and SEO were used - which in the end gave not 40 sales, but 50. This is the synergy effect: a situation where the interaction of two or more factors gives an increase in the result more than each of these factors could give by separately.

Using two or more advertising channels at the same time, we get a great return. Knowing firsthand about the synergy effect, our Internet marketers strive to use the maximum advertising channels. We recommend that you take note of such a little trick :) Now let's move on to a specific example that will illustrate all of the above - a case on the "paid traffic" service in the subject of tires.

I’ll immediately attach a fresh screenshot from Google Analytics, as I know that case readers love them very much:

This case reflects the further results of the project, the case of which I posted last year. Compare these two years. To begin with, let's compare the expenses of each season - 2012 and 2013 (by season I mean the period from 1.09. to 31.12):

• advertising in price aggregators;
• contextual advertising.

In the 2012 season, advertising was used in Google Ads and placement on two price lists: Yandex.Market and Hotline.ua. In the same season of 2013, advertising was already used in Google Ads, Yandex.Direct and on 10 price aggregators. The use of additional advertising channels increased costs by almost 310%. Now let's see how, with an increase in advertising costs by 310%, the income from the project increased:

Thus, we see that by increasing advertising costs by 310%, we increased the client's income not by 310%, but by 573%. Wonderful, isn't it?! That is, the growth of income in comparison with spending is not linear, but exponential.

In obtaining such a result, of course, there was a synergy effect.

Let's look at the growth in gross profit:

Let's also illustrate how the number of transactions has grown:

This screenshot allows you to draw conclusions about the situation with the average bill. If the income grew by 573%, and the number of sales by 557%, then it becomes clear that the average check has increased slightly.

With data on revenue from Google Analytics, spending and margin, we calculate the most important performance indicator - ROMI (return on marketing investment) using the following formula:

ROMI = ((Revenue × Margin) - Customer Spending) / Customer Spending

So let's compare the ROMI results of the two seasons:

It is important to note that when calculating ROMI, we took into account only the income that Google Analytics shows, which means that we did not take into account another 80% of sales that were made by phone, that is, we took into account only 20% of the client's income - this just part 5.

A very interesting situation emerges when we calculate our ROMI with 80% of phone orders. To do this, we multiply our income by 5, and then we count as usual:

The growth of ROMI on a more realistic income looks even more attractive. However, the point is not only in ROMI, but in a real increase in turnover: significantly more customers -> significantly more sales.

Now again the results of the 2013 season

Customer spending: 131 341 UAH (including agency fees). Margin: 13%. Number of transactions: 880. Google Analytics Revenue: UAH 1,317,166.2 Gross profit (including phone orders): 856 158 UAH ROMI by gross margin (including phone orders) : 551,86%.

Of course, the result obtained is far from the limit: there is room to grow the advertising budget > there is room to grow the client's income. Next season, we will definitely use additional advertising channels (their number will probably never end).

Among the must-have features of the new season is the use of the ifTheyCall phone order tracking tool. This is a novelty from Netpeak, which we simply did not have time to use in the September-December 2013 season. This tool will allow you to more accurately assess the return on each advertising channel, redistribute the budget and be even more efficient.

I will illustrate the results in the form of pictures

As you can see from the chart, the breakeven point is at the bottom. Up to this point, the investment in advertising will not pay off. For example, if you spend 100 UAH. to get 100 clicks - the probability of getting a sale that would pay off these investments is almost 0. The second point on the graph is the optimum point (let's call it that) - this is when you invest the maximum in advertising and get the maximum income. After this point, saturation is observed, that is, the market is saturated, advertising covers all potential buyers, the growth of investments in advertising does not give more income growth. If your advertising budget is below the break-even point, then chances are that by investing twice as much in advertising, your income will grow exponentially until the optimum point is reached.

• synergy effect from the use of 2 or more advertising channels at the same time:

It only remains to add to this illustration - try new advertising channels :)

As already emphasized in the previous section, any population is, in principle, capable of exponential growth, and this is why the exponential model is used to estimate the potential for population growth. In some cases, however, the exponential model turns out to be suitable for describing actually observed processes as well. Obviously, this is possible when for a sufficiently long (relative to the duration of the generation) time, nothing limits the growth of the population and, accordingly, the indicator of its specific rate ( r) keeps a constant positive value.

So, for example, in 1937, 2 males and 6 females of pheasant were brought to the small island of Protekshi (off the northwestern coast of the USA near Washington State). (Phasanius colchicus torqualus), never seen before on the island. In the same year, pheasants began to breed, and after 6 years, the population, which began with 8 birds, already consisted of 1898 individuals. As follows from Fig. 28 A, during at least the first 3-4 years, the growth in the number of pheasants was well described by an exponential dependence (a straight line with a logarithmic scale along the ordinate). Unfortunately, later, due to the outbreak of hostilities, troops were stationed on the island, annual counts ceased, and the pheasant population itself was largely exterminated.

Another famous case exponential population growth - an increase in the population of the ringed dove (Streptopelia decaocto) in the British Isles in the late 1950s and early 1960s. (Fig. 28, b). This growth stopped only 8 years later, after all suitable habitats were settled.

The list of examples of exponential population growth could be continued. In particular, several times the exponential (or at least close to exponential) increase in the number of reindeer (Rangifer tarandus) was observed during its introduction to various islands. So, from 25 individuals (4 males and 21 females), brought in 1911 to St. Paul Island (included in the archipelago of the Pribylov Islands in the Bering Sea), a population arose, the number of which by 1938 was 100%. reached 2 thousand individuals, but then a sharp decline followed, and by 1950 only 8 deer remained on the island. A similar pattern was observed on the island of St. Matthew (also located in the Bering Sea): 29 individuals (5 males and 24 females) introduced to the island in 1944, gave a population of 1350 individuals in 1957, and in 1963 - about 6 thousand individuals (the area of ​​\u200b\u200bthis island is 332 km 2, which is approximately three times the area of ​​St. Paul's Island). In subsequent years, however, there was a catastrophic decrease in the number of deer - by 1966 there were only 42 of them left. In both cases described above, the cause sharp decline there was a shortage of food in winter, consisting almost exclusively of lichens.

In the laboratory, conditions for exponential growth can be created by supplying cultivated organisms with an excess of resources that usually limit their development, and also by maintaining the value of all physicochemical parameters of the environment within the tolerance of a given species. Often, in order to maintain exponential growth, it is necessary to remove metabolic products of organisms (using, for example, flow systems when cultivating various aquatic animals and plants) or isolate nascent individuals from each other in order to avoid their crowding (this is important, for example, when cultivating many rodents and other animals with enough complex behavior). In practice, it is not difficult to obtain an exponential growth curve in an experiment only for very small organisms (yeast fungi, protozoa, unicellular algae, etc.). large organisms cultivate in large quantities difficult for purely technical reasons. In addition, this takes a lot of time.

Situations in which the conditions for exponential growth are formed are also possible in nature, and not only for island populations. So, for example, in lakes temperate latitudes in the spring, after the ice melts, the surface layers contain a large number of usually deficient for planktonic algae of biogenic elements (phosphorus, nitrogen, silicon), and therefore it is not surprising that immediately after the water warms up, there is a rapid (close to exponential) increase in the number of diatoms or green algae. It stops only when all the deficient elements are bound in the cells of algae or when the production of populations is balanced by their consumption by various phytophagous animals.

Although other examples of actually observed exponential increases in numbers can be cited, they cannot be said to be very numerous. Obviously, an increase in the population according to an exponential law, if it occurs, is only very a short time, followed by a decline or a plateau (= stationary level). In principle, there are several options for stopping the exponential growth in numbers. The first option is the alternation of periods of exponential growth in numbers with periods of sharp (catastrophic) decline, down to very low values. Such regulation (and by population regulation we mean the action of any mechanisms leading to population growth restriction) is most likely in organisms with a short life cycle living in places with pronounced fluctuations in the main limiting factors, for example, in insects living in high latitudes. It is also obvious that such organisms must have dormant stages that allow them to survive unfavorable seasons. The second option is to abruptly stop the exponential growth and maintain the population at a constant (=stationary) level, around which various fluctuations are possible. The third option is a smooth exit to the plateau. The resulting S-shaped curve indicates that as the population increases, the growth rate does not remain constant, but decreases. S-shaped population growth is observed very often both in laboratory experiments and when species are introduced into new habitats.

People are not very good predictors of the future. For most of history, our experiences have been “local and linear”: we used the same tools, ate the same foods, lived in a certain place. As a result, our predictive abilities are based on intuition and past experience. It's like a ladder: after taking a few steps up, we understand what the rest of the way along this ladder will be. As we live our lives, we expect each new day to be like the previous one. However, things are changing now.

Renowned American inventor and futurist Ray Kurzweil, in his book The Singularity Is Near, writes that the leap in technology development that we have seen in recent decades has accelerated progress in many different areas. This has led to unexpected technological and social change occurring not only between generations, but also within them. Now the intuitive approach to predicting the future does not work. The future is no longer unfolding linearly, but exponentially: it is increasingly difficult to predict what will happen next and when it will happen. The pace of technological progress constantly surprises us, and in order to keep up with them and learn how to predict the future, you must first learn to think exponentially.

What is exponential growth?

Unlike linear growth, which is the result of repeatedly adding a constant, exponential growth is a multiple multiplication. If linear growth is a straight line stable over time, then the exponential growth line is like a takeoff. How greater value takes on a value, the faster it grows further.

Imagine that you are walking down the road, and each step you take is a meter long. You take six steps and now you have moved six meters. After you take another 24 steps, you will be 30 meters from where you started. This is linear growth.

Now imagine (although your body does not know how, but imagine) that each time the length of your step doubles. That is, first you step one meter, then two, then four, then eight, and so on. In six such steps you will overcome 32 meters - this is much more than in six steps of one meter. It's hard to believe, but if you continue at the same pace, then after the thirtieth step you will find yourself at a distance of a billion meters from the starting point. That's 26 trips around the Earth. And this is exponential growth.

Interestingly, each new step with such growth, this is the sum of all the previous ones. That is, after 29 steps you have overcome 500 million meters, and you overcome the same amount in one next, thirtieth step. This means that any of your previous steps are incomparably small in relation to the next few steps of explosive growth, and most of it occurs within a relatively short period of time. If we imagine such growth as a movement from point A to point B, the greatest progress in movement will be made at the last stage.

We often miss telling trends on early stages, since the initial rate of exponential growth is slow and gradual, it is difficult to distinguish it from linear growth. In addition, often predictions based on the assumption that some phenomenon will develop exponentially can seem incredible, and we refuse them.

“When the scanning of the human genome began in 1990, critics noted that, given the speed at which this process initially proceeded, the genome could only be scanned thousands of years later. However, the project was completed already in 2003,”- Raymond Kurzweil gives an example.

IN Lately Technology development is exponential: every decade, every year we are able to incomparably more than before.

Can exponential growth ever end?

In practice, exponential trends do not last forever. However, some of them can continue for long periods of time if there are appropriate conditions for explosive development.

Typically, an exponential trend consists of a series of successive S-shaped technological life cycles or S-curves. Each curve looks like an "S" because of the three growth stages it shows: initial slow growth, explosive growth, and flattening out as the technology matures. These S-curves intersect, and when one technology slows down, a new one begins to rise. With each new S-shaped loop of development, the amount of time needed to achieve more high levels performance gets smaller.

For example, speaking about the development of technology in the last century, Kurzweil lists five computing paradigms: electromechanical, relays, vacuum tubes, discrete transistors, and integrated circuits. When one technology exhausted its potential, the next one began to progress, and it did it faster than its predecessors.

Planning for an exponential future

In the context of exponential development, it is very difficult to predict what awaits us in the future. Build a graph based on geometric progression- this is one thing, but to estimate how life will change in ten to twenty years is quite another. But you can follow a simple rule of thumb: expect life to surprise you a lot, and plan for the surprises you expect. In other words, you can assume the most incredible outcomes and prepare for them, as if they had definitely taken place.

“The future will be much more amazing than most people can imagine. Only a few have truly grasped the fact that the rate of change itself is accelerating.”- Raymond Kurzweil writes.

What will our life look like in the next five years? One way to make a prediction is to look at the last five years and carry that experience over to the next five, but that's "linear" thinking, which we've found doesn't always work. The pace of change is changing, so the progress that has been made in the past five years will take longer in the future. It is likely that the changes that you expect in five years will actually occur in three or two years. With a little practice, we will be better able to predict the future development of life, we will learn to see the prospects for exponential growth, and we will be better able to plan our own future.

It's not just an interesting concept. Our thinking, sharpened more often for linear development, can lead us to a dead end. It is linear thinking that makes some businessmen and politicians resist change, they simply do not understand that development is exponential, and they worry that it is becoming increasingly difficult to control the future. But this is the field for competition. To keep up with this change, you must always be one step ahead and do not what is relevant now, but what will be relevant and in demand in the future, taking into account that development is not linear, but exponential.

Exponential thinking reduces the destructive stresses that arise from our fear of the future and opens up new possibilities. If we can better plan our future and can think exponentially, we will ease the transition from one paradigm to another and face the future calmly.

Hello! Today we will try to figure out what exponential growth is. Exponential growth - an increase in a value exponentially. The value grows at a rate proportional to its value. This means that for any exponentially growing value, the larger the value it takes, the faster it grows. Let's look at this with an example. You may remember from biology that bacteria multiply VERY fast. The growth of a bacterial population is analogous to the growth of continuously accrued interest. I will show it when we solve the problem. So, this is our task for exponential growth. Here is the condition: initial stage bacterial colony contains 100 cells, and it begins to grow in proportion to its size. After 1 hour, the number of cells increases to 420. First, we need to find an expression that shows the number of bacteria after t hours. Let's get on with this. The number of bacteria is, one might say, a function of time. Let's call it b. So let's write it down. The number of bacteria as a function of t can be written as b(t). I'll write it down here: b(t). Thus, the number of bacteria as a function of time is equal to: the initial number of bacteria, that is, I is zero (if we draw an analogy with interest, then this is our loan body). In this case, this is the number we start with. Next we have number goes e to the power of kt, where k is a form of exponential growth. We have I zero, in other words, the initial quantity. t=0, because at the initial moment of time, time is equal to zero, which means that the entire power is equal to zero, and the entire expression here is equal to one. Logical, right? b(0) must be equal to I zero. Therefore, if you know which value to start with, as well as the second value, then you can find k. Then you substitute the found value instead of k - and now you have completed the first item of the task: find an expression that shows the number of bacteria after t hours. So my question is, what is I equal to zero? We know this number. Here in the problem: at the initial stage, a bacterial colony contains 100 cells. Therefore, we know that b(0) is equal to 100. Let me put it another way: b(0)=I zero*e to the power of 0 =I zero. Therefore, the number of bacteria at t=0 is 100. Here we are a little advanced in the solution. Now we can say that b(t)=100*e to the power of kt. Thus, if we had k, then we could complete the first part of the task: find an expression that shows the number of bacteria after t hours. But how can we find k? And here we have the second value of the number of bacteria: after 1 hour, the number of cells increases to 420 pieces. What does this tell us? That b(1) i.e. the population after 1 hour is 420, or it is equal to 100*e to the power of kt. What is t? t=1, therefore, multiply by e to the power of k. So 420=100*e to the power of k. Now we can find k. Let's start by dividing both sides of the equation by 100. So 4.2... I'll probably swap the parts of the equation. So e to the power of k is 4.2. Now, to find k, we need to take the natural logarithms of both sides. Thus, k=ln(4,2). As a result, we will get some number. We will find it later with a calculator. So, we first substituted the value 100 into this expression, found out what I equals zero and with the help of additional data we found k: k=ln(4,2). Now we have an expression, since k and I are known to be zero. Therefore, here is the answer to the first item of the assignment: the function b (t) is equal to: the initial amount, that is, 100, multiply by e to the power of kt, and since k \u003d ln (4,2), we get e to the power (ln (4 ,2))*t. This is what our function looks like. Now let's move on to the second part of our task. Here it is, the second point: find the number of bacteria after 3 hours. This is easy and simple to do. We have a function, and t=3, so we can find the number of bacteria after 3 hours. So b(3)=100*e to the power of (ln(4,2)*3). And we can calculate the value of this expression, if, of course, you have a calculator. What is the natural logarithm of 4.2? Actually, we can find the value analytically. So, this is the same as 100 times e to the power of ln(4,2) and all this to the third power, because if two powers are multiplied, then this is tantamount to raising to a power, which means we raise to the 3rd power . And if we simplify here, then everything is clear further. But what is e to the power of ln(4,2)? That's 4.2, right? The natural logarithm tells us how much e must be raised to get 4.2. Look, I can even do without a calculator. So 100*(4,2) to the third power. And now we need to find out how much (4,2) will be in the third degree. It will be around 70. Let's deal with this later. Here is the answer to the second paragraph of our task. And you can find the value using a calculator. You can do it yourself. What is the third point? Now we need to find the growth rate after 3 hours. What do they want from us at this point? We need to find the slope of this function. In other words, we need to find the derivative of this function at t=3. Let me delete everything here, since we have already completed these tasks. All you need to do here is use a calculator. Ready. So, let's move on to the third point. We need to find the growth rate, that is, the derivative of this function. So, the derivative of the function b’(t) is… What is it equal to? Let's use the chain rule, i.e. the principle of differentiation of a complex function. So, since 100 is a constant, we can write 100 before the function. And the derivative of this expression is equal to ln(4,2) times the derivative e to the power of ln(4,2)*t. It was we who found the growth rate at t, and we need to find out what it will be equal to at t=3. Therefore, b'(3)=100*ln(4,2), and we multiply all this by e to the power of ln(4,2)*t. And we have already said that this expression is simply equal to (4,2) to the power of t. So here we multiply by (4.2) to the third power. As you can see, we also touched on the topic of logarithms here. Well, then everything is easy and simple: we substituted the value 3 instead of t. I hope you understand. Well, if not, you can simply use the calculator. But, in my opinion, it is necessary to know: e vtepeni (ln x) = x. After all, what is (ln x)? This is the power you need to raise e to get x. In other words, if I raise e to the x power, I get x. That's all I wanted to say. So, e to the power of ((ln(4,2) to the power of t)= (4,2) to the power of t. As you can see, I can rewrite our original expression as follows: 100*(4,2) to the power of t. We just simplified the answer for the first item of the problem, it will be better that way, it would make it easier to find a solution for the second item, but for the third item, it's better to leave everything as it is, since it is much easier to find the derivative of this expression. We can rewrite this expression like this: b'(t)=(100*ln(4,2))*(4,2) to the power of t. So I just changed this expression to this. Sorry, I'm here And finally, we come to the last point of our task: find the time after which the number of bacteria reaches 10,000. Let me probably erase the solution to the third point. After what time will the number of bacteria reach 10,000? Let's first write down our the expression is a bit simpler, so b(t)=100*e to the power of (ln(4,2)*t), which is, as I said, 100*(4,2)^t. We are asked when the number of bacteria will reach 10,000. In other words, at what value of t is the function b(t) equal to 10.000. So 10.000=100*e to the ln(4,2)*t power. Let's see what we have here. We can divide both sides of the equation by 100. Therefore, 100=e to the power of (ln(4,2)*t). And now we can write both parts as natural logarithms. What can we do here? Let's take another color, ln100 is..., and if we take the natural logarithm of e to some power, then we get just the natural logarithm of the value of this power. In other words, we are left with only the logarithm of the expression that is in the power. So let's write it down: ln100=ln(4,2)*t. And to find t, we need to divide both sides of the equation by ln(4,2). Therefore, t=(ln100)/(ln(4,2)) In this way we will find the time after which the number of bacteria reaches 10.000. It remains only to take a calculator and find the value of this expression. And let's now for the sake of interest, consider a simplified version of our expression. So, what would we get: 100*(4,2) to the power of t=10.000. We divide both sides of the equality by 100. Hence, (4.2) to the power of t=100. And to solve this, we need to take the logarithm to base 4.2. Therefore, t is equal to the logarithm of 100 base 4.2. We will come back to this in a video about the properties of the logarithm. It is very important to know how you can calculate the logarithm to the base of some number. Since on a calculator you can only find the logarithm to the base e or 10. But how to find the logarithm to the base of any other number? My answer is very simple: you just need to take the natural logarithm of 100 and divide it by the natural logarithm of this value. Or the decimal logarithm is 100 and divided by the decimal logarithm of 4.2. That's it, we'll probably finish on this, so that everything is not messed up in your head. So, in this lesson, we looked at exponential growth. Instead of "bacterial colony" we could write "the initial deposit is 100 and grows in proportion to its size." Then it would be compound interest. And here we could say that “After 1 hour, the amount has increased by, say, $4.2. In such a case, we would be looking for continuously accrued interest. In general, it's the same. It doesn't matter what we're looking at. In the future, I will show a few more examples on this topic, and we will also consider the problem of exponential decay. See you soon!