Logistic Growth: Overshoot and Collapse

Summary

This lesson covers overshoot and collapse in logistic growth models. It explains how populations typically grow towards a carrying capacity but can sometimes exceed it, leading to overshoot. Through examples, the video demonstrates how populations can fluctuate above and below this limit, sometimes stabilizing, but also leading to extreme behaviors like two-cycle and four-cycle patterns. If overshoot is excessive, it can degrade the environment's capacity, resulting in a population collapse. The lesson highlights real-world instances, like lizard populations, where these dynamics have been observed.

Takeaways

• 📈 Logistic growth involves a population growing towards a carrying capacity, which limits its growth.
• 🔢 The carrying capacity in the example is 1,500, with the growth rate adjusting as the population nears this limit.
• 📊 When the growth rate is high, populations can overshoot the carrying capacity, temporarily exceeding it.
• 📉 After overshooting, populations tend to drop back below the carrying capacity, but the population may still stabilize near it over time.
• 🔄 Overshooting can lead to fluctuating population levels, bouncing above and below the carrying capacity in cycles.
• 🔍 A two-cycle population oscillates between a higher and lower value on either side of the carrying capacity, as seen in lizard populations studied in California.
• 🚀 As the growth rate increases, population behavior becomes more extreme, potentially leading to four-cycle or even more complex population cycles.
• 📉 In extreme cases, overshooting the carrying capacity too much can degrade the environment, leading to a population collapse.
• ⚠️ A collapse is a dramatic decrease in the population following an overshoot that severely exceeds the carrying capacity.
• 📚 Cyclical and collapse behaviors are not just mathematical oddities but have been observed in real-world ecological systems.

Q & A

What is the concept of 'overshoot' in logistic growth?

Overshoot occurs when the population exceeds the carrying capacity due to a high growth rate. This can lead to the population temporarily surpassing the environment's sustainable limit.

What is the carrying capacity in the context of logistic growth models?

The carrying capacity is the maximum population size that an environment can sustainably support. It acts as a limiting factor for population growth.

How does the growth rate affect logistic growth and the potential for overshoot?

A high growth rate can push the population beyond the carrying capacity, causing overshoot. This happens because the population grows too rapidly for the environment to support.

What is a 'two cycle' in population dynamics?

A two cycle occurs when the population alternates between two levels: one above and one below the carrying capacity. This pattern can continue over time, reflecting periodic fluctuations in population size.

Can overshoot lead to long-term consequences for a population?

Yes, significant overshoot can degrade the carrying capacity, resulting in a population collapse. This is a dramatic decrease in population size due to unsustainable environmental conditions.

What is the 'recursive logistic growth equation' mentioned in the script?

The recursive logistic growth equation models population changes over time, accounting for both growth rate and carrying capacity. It helps predict future population sizes based on current values.

What happens when the initial population is close to the carrying capacity?

If the initial population is close to the carrying capacity, the growth rate may cause the population to overshoot slightly before stabilizing near the carrying capacity, as seen in the second example in the script.

What is meant by 'collapse' in the context of population dynamics?

Collapse refers to a severe decline in population size following a substantial overshoot. It happens when the environment's carrying capacity is significantly degraded, no longer supporting the previous population level.

How do four cycles and other higher-order cycles relate to logistic growth?

Higher-order cycles, like four cycles, occur when the population fluctuates among more than two levels. These complex patterns arise when the growth rate is extremely high, causing unpredictable oscillations.

Can you provide a real-world example of a two-cycle pattern in nature?

Yes, researchers from the University of California observed a stable two-cycle pattern in a lizard population in California, where the population alternated between two distinct levels over time.

Outlines

00:00

📈 Introduction to Logistic Growth and Overshoot

This paragraph introduces the concept of overshoot in logistic growth models. It explains how logistic growth has a carrying capacity, often behaving predictably until the population approaches the carrying capacity. However, with a high growth rate, the population can surpass this capacity, resulting in overshoot. The example provided starts with a population of 75 and a 60% growth rate per year, showcasing how the population approaches but does not exceed the carrying capacity of 1,500.

05:00

📊 Example of Overshoot in Population Growth

In this second example, the initial population starts at 1,200 with a much higher growth rate of 160%. This leads to a scenario where the population exceeds the carrying capacity of 1,500 multiple times, an occurrence known as overshoot. After each overshoot, the population dips below the capacity, eventually stabilizing around the carrying capacity. The graph visually represents this overshoot and the subsequent correction as the population approaches equilibrium.

🔄 Two-Cycle Behavior in Population Growth

This paragraph explores a third example where the initial population is 1,100, with an even higher growth rate of 220%. In this scenario, the population repeatedly overshoots and drops below the carrying capacity in a pattern known as a 'two-cycle.' The population alternates between a high and low level around the carrying capacity. This behavior, though seemingly mathematical, has been observed in nature, such as in a lizard population studied by researchers from the University of California. More extreme cycles can occur with higher growth rates.

🔢 Four-Cycle Behavior and Population Collapse

This section highlights more extreme population behaviors, including a four-cycle pattern and cases with no discernible pattern between high and low populations. It introduces the concept of population collapse, which occurs when an overshoot is too large, leading to a degradation of the carrying capacity. As a result, the population may experience a dramatic decline, as shown in the graph where overshoot leads to a sharp drop, representing the collapse.

Keywords

💡Overshoot

Overshoot refers to a situation where the population exceeds the carrying capacity. In the video, overshoot occurs when the growth rate is very high, causing the population to temporarily rise above the capacity, like in the example where the population goes beyond 1,500. This concept highlights the instability that can occur when limits are surpassed.

💡Collapse

Collapse is a dramatic decrease in population that happens when an overshoot severely degrades the carrying capacity. The video explains that if the population grows too far beyond the carrying capacity, the environment can't sustain it, leading to a collapse, as shown in the graph where population drops sharply after overshoot.

💡Carrying Capacity

Carrying capacity is the maximum population size that an environment can sustainably support. In the video, it is consistently set at 1,500 for the examples. Logistic growth models are built around the concept that population growth slows as it nears this capacity.

💡Logistic Growth

Logistic growth describes a population model where growth initially rises rapidly but slows as it approaches the carrying capacity. The video shows a typical logistic growth curve where the population increases and then stabilizes near the capacity of 1,500. This model contrasts with exponential growth.

💡Recursive Equation

A recursive equation is used in the video to calculate the population at various time points, taking into account the growth rate and carrying capacity. It is applied repeatedly to show how the population evolves over time under different conditions, such as varying initial populations or growth rates.

💡Growth Rate

Growth rate refers to how quickly a population increases over time. The video presents examples with different growth rates—like 60%, 160%, and 220%—showing how higher growth rates can push populations beyond the carrying capacity, leading to overshoot or more extreme behaviors like cycles.

💡Two Cycle

A two cycle occurs when a population alternates between levels above and below the carrying capacity without stabilizing. The video describes this phenomenon in one example, where population levels repeatedly oscillate around the carrying capacity, showing instability in population dynamics.

💡Four Cycle

A four cycle is a more extreme form of population fluctuation, where the population cycles through four different levels. In the video, this concept is introduced to show how higher growth rates can lead to increasingly complex and less predictable population dynamics, as demonstrated in a four-cycle example.

💡Initial Population

Initial population refers to the starting number of individuals in the population model. The video uses different initial populations—like 75, 1,200, and 1,100—in its examples to show how varying starting points interact with growth rates and carrying capacities to produce different outcomes, including overshoot and collapse.

💡Population Stability

Population stability occurs when the population remains near the carrying capacity without extreme fluctuations. The video contrasts stable logistic growth, where populations stabilize at or near 1,500, with unstable outcomes like overshoot or cycles, highlighting how growth rate and initial population affect stability.

Highlights

Introduction to the concept of overshoot and collapse in logistic growth models.

Logistic growth models include a carrying capacity that limits population growth.

The carrying capacity for the first example is set at 1,500, with an initial population of 75 and a 60% annual growth rate.

Overshoot occurs when the population exceeds the carrying capacity due to a high growth rate.

In the first example, the population gradually approaches the carrying capacity and stabilizes near it.

In the second example, the initial population is 1,200, with a growth rate of 160%, leading to overshoot.

Overshoot causes the population to exceed the carrying capacity, but it eventually returns to settle near the limit.

Analysis of overshoot using tables and graphs shows populations that exceed and then drop below the carrying capacity.

A two-cycle pattern can emerge, where population levels alternate between higher and lower values.

The two-cycle pattern has been observed in real-world populations, such as in a lizard population in California.

Higher growth rates can lead to more extreme behaviors, like stable four cycles and eight cycles.

An example of a four-cycle pattern demonstrates alternating high and low population levels.

Extremely high overshoot can degrade the carrying capacity, leading to a population collapse.

A collapse is a severe drop in population, caused by overshoot that significantly exceeds the carrying capacity.

Conclusion: A review of overshoot, two-cycle, four-cycle, and population collapse, emphasizing the real-world application of these patterns.

Transcripts

- WELCOME TO A LESSON ON OVERSHOOT AND COLLAPSE

INVOLVING LOGISTIC GROWTH.

WE OBSERVED IN THE PREVIOUS LESSON

THAT LOGISTIC GROWTH MODELS HAVE A CARRYING CAPACITY.

WE SAW THAT LOGISTIC GROWTH OFTEN BEHAVES LIKE THE GRAPH

SHOWN HERE BELOW.

NOTICE HOW IT APPEARS

THE CARRYING CAPACITY WOULD BE 1,500.

THE GROWTH RATE ADJUSTS

AS THE POPULATION APPROACHES THE CARRYING CAPACITY.

HOWEVER, IF THE GROWTH RATE IS VERY HIGH,

IT CAN MOVE THE POPULATION BEYOND THE CARRYING CAPACITY.

THIS IS CALLED OVERSHOOT.

LET'S TAKE A LOOK AT SEVERAL EXAMPLES.

FOR THE POPULATION BELOW THE CARRYING CAPACITY IS 1,500,

THE INITIAL POPULATION WAS 75, AND ABSENT RESTRICTIONS,

THE GROWTH RATE WOULD BE 60% PER YEAR.

SO USING OUR RECURSIVE LOGISTIC GROWTH EQUATION HERE,

FROM OUR PREVIOUS LESSON WE SHOULD BE ABLE TO PERFORM

A SUBSTITUTION FOR R AND K,

AND THEN FIND THE POPULATION LEVELS GIVEN IN THE TABLE.

YOU MAY WANT TO PAUSE THE VIDEO

JUST TO VERIFY YOU CAN FIND THESE POPULATION LEVELS.

ANALYZING THE TABLE,

NOTICE HOW THE POPULATION GRADUALLY INCREASES.

THEN LOOKING AT THE GRAPH HERE THAT HAS ADDITIONAL VALUES,

WE CAN SEE THE POPULATION INCREASES GRADUALLY

AND THEN SLOWS AS IT APPROACHES THE CARRYING CAPACITY OF 1,500.

AGAIN, HERE'S THE CARRYING CAPACITY,

AND THIS GRAPH BEHAVES GENERALLY LIKE EXPECTED

WHEN WE HAVE LOGISTIC GROWTH.

BUT, AGAIN, AS JUST MENTIONED, THIS IS NOT ALWAYS THE CASE.

SO LET'S TAKE A LOOK AT A SECOND EXAMPLE.

FOR THE POPULATION BELOW, AGAIN, THE CARRYING CAPACITY IS 1,500,

BUT IN THIS CASE THE INITIAL POPULATION WAS 1,200

AND ABSENT ANY RESTRICTIONS THE GROWTH RATE WOULD BE 160%.

SO OF COURSE THIS WILL AFFECT

THE RECURSIVE LOGISTIC GROWTH EQUATION GIVEN HERE,

WHICH WE SEE HERE BELOW WITH THE GROWTH RATE

AND CARRYING CAPACITY SUBSTITUTED INTO THE EQUATION.

LET'S BEGIN BY LOOKING AT THE TABLE OF VALUES.

NOTICE HOW P SUB 1, P SUB 3, AND P SUB 5

ALL HAVE POPULATION LEVELS ABOVE THE CARRYING CAPACITY OF 1,500.

THIS IS OVERSHOOT.

BUT NOTICE HOW THE YEAR AFTER THE OVERSHOOT

THE POPULATION GOES BACK DOWN BELOW THE CARRYING CAPACITY.

BUT IT DOES APPEAR OVER TIME THE POPULATION IS GOING TO SETTLE

NEAR THE CARRYING CAPACITY OF 1,500.

LOOKING AT THE GRAPH, AGAIN, HERE IS THE CARRYING CAPACITY,

THE POINTS ABOVE THIS HORIZONTAL LINE REPRESENT OVERSHOOT,

AND OUR GRAPH WOULD LOOK LIKE THIS.

NOT THE TRADITIONAL GRAPH WE WOULD EXPECT

WHEN WORKING WITH LOGISTIC GROWTH.

NOW, LET'S TAKE A LOOK AT ONE MORE EXAMPLE.

AGAIN, THE CARRYING CAPACITY IS 1,500,

THE INITIAL POPULATION WAS 1,100,

BUT ABSENT ANY RESTRICTIONS THE GROWTH RATE WOULD BE 220%.

SO, AGAIN, USING OUR RECURSIVE EQUATION HERE BELOW,

WE SHOULD BE ABLE TO GENERATE THE POPULATION LEVELS

GIVEN IN THE TABLE.

ANALYZING THE TABLE,

NOTICE HOW, AGAIN, P SUB 1, P SUB 3, AND P SUB 5

ALL HAVE POPULATION LEVELS ABOVE THE CARRYING CAPACITY OF 1,500,

ANOTHER EXAMPLE OF OVERSHOOT.

BUT ANALYZING THIS MORE CLOSELY, NOTICE HOW IT DOES APPEAR

THAT THE POPULATION LEVELS ARE BOUNCING BACK AND FORTH

BETWEEN ONE POPULATION LEVEL ABOVE THE CARRYING CAPACITY

AND ONE POPULATION LEVEL BELOW THE CARRYING CAPACITY.

IF WE PLOT THESE POINTS ON THE COORDINATE PLANE,

IT BECOMES EVEN MORE OBVIOUS.

HERE'S THE CARRYING CAPACITY, OUR GRAPH WOULD LOOK LIKE THIS,

AND THIS IS CALLED A TWO CYCLE.

AGAIN, THIS IS BECAUSE THE POPULATIONS

ARE BOUNCING BACK AND FORTH

BETWEEN THE SAME HIGHER LEVEL POPULATION

AND SAME LOWER LEVEL POPULATION.

WELL, IT MIGHT BE TEMPTING TO TREAT THIS

ONLY AS A STRANGE SIDE AFFECT OF MATHEMATICS.

THIS IS ACTUALLY BEEN OBSERVED IN NATURE.

RESEARCHERS FROM THE UNIVERSITY OF CALIFORNIA

OBSERVED A STABLE TWO CYCLE IN A LIZARD POPULATION IN CALIFORNIA.

TAKING THIS EVEN FURTHER,

WE CAN GET MORE AND MORE EXTREME BEHAVIORS

AS THE GROWTH RATE INCREASES HIGHER.

IT IS POSSIBLE TO GET STABLE FOUR CYCLE, AND EIGHT CYCLES,

AND HIGHER.

HERE'S AN EXAMPLE OF A FOUR CYCLE,

AND HERE'S AN EXAMPLE WHERE THERE SEEMS TO BE NO PATTERN

BETWEEN THE HIGHER POPULATIONS AND LOWER POPULATIONS.

IF A POPULATION OVERSHOOTS THE CARRYING CAPACITY BY TOO MUCH,

THIS CAN DEGRADE THE CARRYING CAPACITY

AND LEAD TO A SEVERE DECREASE OF THE POPULATION.

THIS IS CALLED A COLLAPSE.

SO LOOKING AT THE GRAPH HERE BELOW, HERE'S THE OVERSHOOT.

AND BECAUSE THE OVERSHOOT IS SO HIGH

ABOVE THE CARRYING CAPACITY,

NOTICE HOW THIS DEGRADES THE CARRYING CAPACITY

AND RESULTS IN A COLLAPSE,

WHICH IS A DRAMATIC DECREASE IN THE POPULATION,

WHICH WE SEE HERE.

THAT'S IT FOR THIS LESSON.

I HOPE YOU FOUND THIS HELPFUL.