The Logistic Model | Free Calculus 2 Lesson (4 College Credits) | UPI Study

UPI Study › Courses › Calculus 2 › Free lessons › Lesson 5.5 Free lesson

The Logistic Model

Calculus 2 · Chapter 5: Differential EquationsSection 5.5 · lesson 26 of 549 min · free

← Exponential Growth and DecayNext lesson: Sequences →

Earn 4 college credits for Calculus 2Recommended for credit by NCCRS · Math 265 · self-paced · one payment, lifetime access
Enroll in Calculus 2 · $250 →Download the syllabus (PDF)

In this lesson

Growth with a ceiling.

  • Read the logistic factor as a brake on exponential growth
  • Find the two equilibria and describe their stability
  • Solve the equation by separation and partial fractions
  • Locate the inflection point and interpret the S-curve

Transcript

Everything said in the video, section by section (1,314 words). Read along, or search it.

MAT-265 - Calculus 2 · Section 5.5

The Logistic Model

The exponential model from the last section predicts a population racing to infinity. Which is useful — but only while resources appear unlimited. Real environments don't work that way. Every habitat has a carrying capacity. The largest population it can sustain. Food runs out. Space runs out. Waste accumulates. So the logistic model adds a single factor that encodes that ceiling. And the remarkable thing is how much that one factor changes.

The Brake

One Extra Factor Changes Everything

Here's the equation. d P d t equals k P, times the quantity one minus P over M. Where M is the carrying capacity. Compare that to the exponential model, d P d t equals k P. The only difference is that extra factor. So let's understand exactly what it does. When P is small relative to M, the ratio P over M is nearly zero. So the factor is close to one, and the equation behaves like d P d t equals k P. Ordinary exponential growth. When P approaches M, the ratio approaches one. So the factor approaches zero, and growth nearly stops. And if P ever exceeds M — say through a temporary overshoot — the factor turns negative. So d P d t is negative, and the population declines back toward M. In every single case, the population is steered toward M. That one factor converts runaway exponential growth into self-limiting growth.

Equilibria

Two Constant Solutions, With Opposite Characters

Before solving, let's find the constant solutions. An equilibrium solution is a population that does not change. One where d P d t equals zero. Set the right side of the logistic equation to zero. k P, times one minus P over M, equals zero. A product is zero when either factor is zero. So there are two solutions. P equals zero. Or one minus P over M equals zero, which gives P equals M. Now here's the interesting part. Those two equilibria have opposite characters. P equals zero is unstable. A population of exactly zero stays at zero — nothing to reproduce. But the slightest positive population grows away from it. P equals M is stable. Populations below M rise toward it. Populations above M fall toward it. So M acts as an attractor. And that matches the intuition perfectly. An environment's carrying capacity is the level a population naturally settles at.

Solving It

Separation, Then Partial Fractions

The logistic equation is separable. So it can be solved explicitly. Separate the variables — all P terms on one side, all t terms on the other. You get d P over P times one minus P over M, equals k d t. Now the left side requires partial fractions. Which is exactly what section one point five was for. Rewrite one over P times one minus P over M, as M over P times M minus P. And that decomposes into one over P, plus one over M minus P. You can verify it: combining those over the common denominator gives back M over P times M minus P. Integrate both sides. You get log absolute P, minus log absolute M minus P, equals k t plus C. For a population between zero and M, both P and M minus P are positive. So the absolute values drop. Combine the logarithms into log of P over M minus P. Exponentiate, let A naught be e to the C, solve for P algebraically, and you get the standard closed form. P of t equals M, over one plus A e to the minus k t. Where A equals M minus P naught, over P naught. Let's use it. A lake is stocked with four hundred fish. The carrying capacity is ten thousand, and the growth constant is zero point four per year. First compute A. Ten thousand minus four hundred, over four hundred. That's ninety-six hundred over four hundred. Twenty-four. So P of t equals ten thousand, over one plus twenty-four e to the minus zero point four t. At t equals five, the exponential term is e to the minus two. About zero point one three five three. So P at five is ten thousand over four point two four seven. About two thousand three hundred and fifty-four fish. Still well below carrying capacity — in the steep, fast-growing middle phase.

Figure 5.5.1

The S-Shaped Solution Curve

The closed-form solution traces a distinctive S-shape. A sigmoid. And it unfolds in three phases. Early on, with P small, growth is slow but accelerating. The curve is concave up. As P passes the midpoint, M over two, the growth rate peaks and the curve switches to concave down. Finally, as P nears M, growth tapers and the curve flattens — approaching the horizontal asymptote P equals M, without ever quite reaching it. The moment of fastest growth — the inflection point — occurs exactly at P equals M over two. Halfway to capacity. And you can confirm that by maximizing the growth rate itself. d P d t equals k P times one minus P over M. As a function of P, that's a downward-opening parabola, with vertex at P equals M over two. At that population, the colony is adding members faster than at any other time. Before it, growth is still ramping up. After it, the environmental brake dominates. Now look at the red curve, for contrast. That's exponential growth from the same starting point. It shoots straight past M and off the chart. That's the behavior the logistic brake prevents.

Contrast

Exponential Versus Logistic

Let's put the two models side by side. The exponential equation is d P d t equals k P. Its solution is P naught e to the k t. The logistic equation adds the factor. Its solution is M over one plus A e to the minus k t. Long-term behavior: exponential grows without bound. Logistic approaches M. Per-capita rate: exponential holds constant at k. Logistic decreases as P approaches M. Curve shape: exponential is ever-steepening. Logistic is S-shaped, and levels off. The crucial difference is the long-term fate. Exponential growth predicts a population racing to infinity. Useful only while resources appear unlimited. Logistic growth bends that trajectory into a realistic plateau at M. But here's the important nuance. For small P, the two models are nearly indistinguishable. Which is exactly why exponential growth works well in the early stages of an outbreak, a new market, or a fresh culture. The models part ways precisely when the population grows large enough for limits to matter.

Key Takeaways

What to Carry Forward

Here's something worth thinking about. Both models begin with the same factor, k P. Yet one runs to infinity, and the other settles at M. The entire difference is that single term, one minus P over M. Why does subtracting one factor change the destiny of the population so completely? Because it makes the growth rate depend on how full the environment already is. Exponential growth is blind to its surroundings. It grows at rate k whether the population is ten or ten million. The logistic factor gives the population feedback about its own size. And that feedback is what bounds it. Five takeaways. The logistic equation models growth limited by a carrying capacity M, with the extra factor slowing growth as P rises. It has two equilibrium solutions. P equals zero, which is unstable. And P equals M, which is stable. Populations are always steered toward M. Solving by separation of variables and partial fractions gives P of t equals M over one plus A e to the minus k t, with A equal to M minus P naught, over P naught. The solution is an S-shaped curve. Slow start, fastest growth at the inflection point P equals M over two, then leveling off toward the asymptote. And unlike the unbounded exponential model, logistic growth approaches a finite limit. The two agree while P is small, but diverge as the population nears its environmental ceiling. That closes Chapter Five. Next up is Chapter Six — Sequences and the Introduction to Series.

About Calculus 2 at UPI Study

Calculus 2 (Math 265) is a self-paced online course of 54 video lessons in 10 chapters, from integration techniques to series, parametric curves and vectors. It carries a NCCRS credit recommendation for 4 semester credits.

Every lesson video and transcript is free on pages like this one. Enrolling adds a practice quiz after each lesson, the graded work and final exam that earn the credit, lifetime access, and an official transcript on completion. UPI Study has helped over 130,000 students with college credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Enroll now → Course details Download the syllabus Every course with PRO →

Why take it here

Why students take Calculus 2 with UPI Study

Open the door to higher mathCalculus 2 is the gateway to Calculus 3, differential equations, statistics and most engineering, physics, economics and computer-science degrees. Completing it now lets you qualify for those courses at your university.
Credit at a fraction of the priceThe same 4 credits can cost $750 or more at a US college. Here it is one payment, self-paced, with lifetime access.
Finish your degree soonerTransfer credits reduce the courses you still owe, so you can graduate earlier or lighten a heavy semester.
Strengthen an applicationA completed college-level course on an official transcript shows admissions and scholarship committees you can do the work.
Study on your own scheduleNo start dates, no deadlines. Watch the lessons, practice, and take the final when you are ready.
130,000+students helped with college creditsACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Where the credit goes

Universities that work with NCCRS

Calculus 2 carries a NCCRS credit recommendation. These universities work with NCCRS; some are UPI Study's signed partners. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Thomas Edison State UniversityThomas Edison State UniversitySigned partnerUniversity of the PeopleUniversity of the PeopleSigned partnerCharter Oak State CollegeCharter Oak State CollegeSigned partnerStrayer UniversityStrayer UniversitySigned partnerColumbia Southern UniversityColumbia Southern UniversitySigned partnerIndiana TechIndiana TechSigned partnerPenn StatePenn StateWorks with NCCRSSouthern New Hampshire UniversitySouthern New Hampshire UniversityWorks with NCCRSSUNYSUNYWorks with NCCRS +many more cooperating universitiesFind my college →

Find my college → Not listed? Ask Athena in the corner of this page, or book a 15-minute call.

FAQ

Questions students ask us

About The Logistic Model

What does the factor (1 - P/M) do in the logistic growth equation?
The factor 1 - P/M acts as a brake that slows growth as the population P approaches the carrying capacity M. When P is small, the factor is near 1, so dP/dt = kP(1 - P/M) behaves like exponential growth, dP/dt = kP. As P nears M, the factor approaches 0 and growth nearly stops. If P exceeds M, the factor turns negative, so the population declines back toward M.
What are the equilibrium solutions of the logistic equation, and are they stable?
The logistic equation has two equilibria, P = 0 and P = M, found by setting kP(1 - P/M) = 0. The equilibrium P = 0 is unstable, because any small positive population grows away from it. The equilibrium P = M is stable. Populations below M rise toward it and populations above M fall toward it, so the carrying capacity acts as an attractor.
How do you solve the logistic differential equation?
Separate the variables and use partial fractions to get P(t) = \frac{M}{1 + Ae^{-kt}}, where A = \frac{M - P_0}{P_0}. The key step is rewriting \frac{1}{P(1 - P/M)} = \frac{1}{P} + \frac{1}{M - P} before integrating. For example, with P_0 = 400, M = 10000, and k = 0.4, you get A = 24. Then P(5) = \frac{10000}{1 + 24e^{-2}} \approx 2354.
Where is the inflection point of a logistic curve?
The inflection point of a logistic curve occurs at P = M/2, halfway to the carrying capacity, where growth is fastest. As a function of P, the rate dP/dt = kP(1 - P/M) is a downward-opening parabola with its vertex at P = M/2. Below that population, the S-curve is concave up and accelerating. Above it, the curve is concave down and levels off toward the asymptote P = M.

About the Calculus 2 course

How many college credits is Calculus 2?
Calculus 2 (Math 265) carries a NCCRS credit recommendation for 4 semester credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.
How much does Calculus 2 cost and what is included?
$250, paid once, with lifetime access: all 54 video lessons, a practice quiz after each, the graded work and final exam, and an official transcript when you finish. Students who want several courses choose UPI Study PRO instead.
Which universities accept UPI Study credits?
UPI Study has signed partner agreements with universities such as Thomas Edison State University, University of the People and Charter Oak State College, and works with universities such as Penn State, SNHU and SUNY. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities. Use "Find my college" on this page to check yours.
How long does Calculus 2 take?
It is self-paced with no deadlines. The 54 lessons total about 9 hours of video; most students add reading and practice time and finish in a few weeks to a few months, on their own schedule.
How are the exams done?
Quizzes and assignments are online in the course. The final exam is taken online with proctoring; a person on the UPI Study team checks assignments, usually within two days.
How do my credits reach my college?
When the course shows 100% complete, request the official transcript from your dashboard. It is sent to the college you name, and you can send it to more than one. Check your college first with "Find my college" on this page, or ask Athena.

Ready for the full course? Earn 4 credits in Calculus 2.

54 video lessons, a quiz after each one, and the graded work that earns the credit. Recommended for credit by NCCRS.

Enroll in Calculus 2 →Download the syllabusEvery course with PRO →
Calculus 2
4 credits · NCCRSEnroll →