How Do Biological Population Trends Work?

Logistic growth model - Woudloper, via Wikimedia Commons.

The human population has increased dramatically over the last two centuries. Since the Industrial Revolution, our population has grown from one to almost eight billion. This growth can be attributed to improved healthcare, industrialization and an overall higher quality of life, among other factors. For humans, our growth can be modeled by the demographic transition model, which places a nation or country in one of five potential stages:

  1. A pre-industrial society in which there are high birth rates and high death rates, resulting in low – or no – population growth. Not a single country on the planet remains in stage one, although some South American, Asian and African countries recently advanced from this stage.

  2. A newly industrialized society in which there are high birth rates and declining death rates, resulting in exponential population growth. Nigeria is an example of a stage two country.

  3. A mature industrial society in which there are declining birth rates, flattening death rates, and decreasingly rapid population growth. India is an example of a stage three country.

  4. A post-industrial society in which birth rates flatten to be at or below replacement level  – 2.1 to 2.3 children per woman – and death rates incline, thereby further decreasing population growth and, potentially, leading to zero population growth (ZPG). The United States is an example of a stage four country.

  5. A society in which there are exceptionally low birth rates, again below replacement level, and high death rates, leading to population decline. Japan is an example of a stage five society.*

Indeed, the demographic transition model works for us, but not for the millions of other species around the world. Regarding biological population patterns, we need a whole new set of science to explain it. Let us learn about population patterns in the natural world.

Population growth patterns in nature

Before we discuss population dynamics in natural environments, we must first understand the characteristics of population growth and decline. Five important phenomena – exponential growth, logistic growth, carrying capacities, overshoots, and die-offs – are the products of population change in an environment.

Exponential growth is, like the term says, exponential – population growth increases at an increasing rate. The function of exponential growth is f(x) = a(1+r)^x, a being the initial population, r being the rate of growth, and x being the number of time intervals the exponential growth occurs. Unmitigated exponential growth is incredibly rare in the natural world, however. Logistic growth – which incorporates exponential elements as well as other ecological phenomena like carrying capacity – is much more common.

Logistic growth is defined through the equation dNdT=rN(K-NK), where K is the carrying capacity – the greatest number of individuals in a population that the environment can support – r is the maximum growth rate for the population, and N is the number of individuals in the population. When graphed, the differential equation resembles a sideways S, with a low N followed by exponential growth and, later, an oscillating curve around a center K (the carrying capacity). Logistic growth is characterized by an initial low-growth or zero-growth population followed by exponential population growth, which eventually levels off as the population reaches the carrying capacity for its environment.

Carrying capacity and effects

The carrying capacity is quite self-explanatory: it is simply the maximum limit on the number of individuals in a species in an environment. If the population of fig trees in an environment is 73, for example, and the carrying capacity is 100, then the fig tree population could theoretically increase by 27 before reaching the carrying capacity. Once the population transcends the carrying capacity, a bunch of wild things – like overshoots and die-offs – begin to happen.

The carrying capacity, as mentioned earlier, is the maximum population of a species that an environment can support; sometimes, the number of individuals in a population increases exponentially, so much so that its population becomes greater than its carrying capacity. When the population “overshoots” the carrying capacity of its environment, leading to a “die-off.” Once a population overshoots its carrying capacity, the environment no longer has the resources to support it; unless at least some of the population migrates, a large proportion of the population will die, leading the population, once again, below the carrying capacity. Continuous overshoots and die-offs oscillate around a central population level – the carrying capacity – until a significant environmental factor, like disease or predation, breaks the status quo.

Predation and population patterns

Population fluctuations resulting from predation apply to both the predator and the prey – major changes that occur in one level of the food chain affect the entire food chain. To understand the effects of over-predation, let us first imagine a secondary consumer preying on a primary consumer; a secondary consumer is a carnivore that eats small animals, whereas a primary consumer is an herbivore that eats plants. The presence of producers is observed to have the most significant impact on the food chain:

  • If plant numbers are rich in an ecosystem, primary consumers will thrive, resulting in an exponential increase in their population due to the abundance of food. As the primary consumer population increases, so will the secondary consumer population; and if the secondary consumers are at the top of the food chain, then the only hindrances to their population growth will be interspecies competition, intraspecies competition and disease – assuming that the population of the primary consumers never decreases.

  • If plant numbers are poor in an ecosystem, the number of primary consumers will, too, be poor, resulting in either a die-off or in zero population growth for all consumers. The loss of plant life, therefore, reverberates through the entire food chain.

The dynamics of predation lead to changes in populations, both of the predator and the prey; let us consider the Canadian Lynx and the Snowshoe Hare; the Snowshoe Hare is a secondary consumer, whereas the Canada Lynx is a tertiary consumer occupying the top of the food chain. The importance of this relationship is remarkably pronounced, as the Snowshoe Hare is virtually the only food source for the Canada Lynx. The dynamics from the predator-prey relationship resemble best how crucial predation is to population patterns:

When the Snowshoe Hare population is high and the Canada Lynx population is low, there is an abundance of food for the Canada Lynx; therefore, the Canada Lynx population will grow to the point at which predation causes the hare population to fall. 

Consequently, as hare become less available to consume, the Canada Lynx will also die off, allowing the hare population to increase without predation from the lynx. The hare population will increase to the point at which the abundance of hares will allow the lynx population to grow once again, resulting in population growth for the lynx and, eventually, decline for the hare. The process is destined to start all over again once the lynx hunts the hare to endangerment.

As the hare is the lynx’s primary food source, the population of the lynx is inextricably connected to that of the hare. When the hare population increases, the lynx population does as well, and vice versa.

Reproductive patterns and population: R- and K-selection

Reproductive patterns, too, contribute to population growth and decline; there are two types of species – K-selected and R-selected – whose reproductive patterns determine, among other things, the steepness of their curve during exponential growth, how far the population overshoots, and how many individuals die off after the overshoots.

R-Selected species – like mice, many insects, rabbits, weeds, bacteria – produce significant offspring in their lifespan. Their offspring, however, are nurtured very little – most R-selected species leave their offspring shortly after or even before their birth. Most R-selected species are egg-bearing, as the self-sustaining environment of the egg allows the parent to nurture more offspring without supporting another in its body. Most R-selected offspring are abandoned from birth, and their path to adulthood, though short, is extremely difficult – a vast majority of the offspring will die long before adulthood.

K-Selected species – like humans, other large mammals, birds, horses, and large plants – produce limited offspring throughout their lifetime. K-selected species cherish and protect their children until adulthood, leading to a high survival rate for the offspring. K-Selected species generally live much longer than R-Selected species; they also tend to die at old ages, rather than at young or random ages.

R-selected species, because of their vast numbers of offspring, are much more likely to undergo exponential growth; the rate of exponential growth for R-selected species is also much greater than K-selected. An R-Selected population is far more inconsistent than the K-Selected populations are. An R-selected, invasive species – most invasive species, excluding humans, are R-selected – as it is introduced to the environment, will shoot straight to the carrying capacity, and because of its quick overshoot, will die off immediately. The die-offs that an R-Selected population experiences are much larger because the overshoots, too, were much greater than occurs with K-selected species. Because the few remaining will continue to produce thousands of offspring, the population will soon shoot right back up again once the environmental conditions are right. The populations of such species are, therefore, much less consistent.

K-Selected species, because of their smaller numbers of offspring, are less likely to experience rapid population growth. Even if it does experience exponential growth, the population growth is still quite slow; the K-selected species simply cannot support so many offspring at once. As the K-Selected population begins to reach the carrying capacity, its population growth slows, and halts, with small overshoots and die-offs continuing until the inevitable moment that a large die-off restarts the process. Exponential growth to an environmental carrying capacity takes significantly longer for a K-Selected species than for an R-Selected species (for the same reasons), and the population fluctuations for a K-Selected species are significantly less so than for an R-Selected species.

Wrapping it up

Biological population dynamics are significantly different from those of the industrialized human population. While our population dynamics depend on development and crop yields, the natural world’s population dynamics depend exclusively on a population’s ability to find food and reproduce. As always, take care and stay curious, everyone.

* The stage five society in the demographic transition model is a hypothesized stage; while there is evidence that affirms its existence, we have yet to see it at a wider scale – although, as the most advanced European and Asian countries grow older, we may find more evidence in coming decades.

If you have any questions, comments, or corrections, please comment on this post or email with your concerns. Thank you.

Note: As a result of temporary changes to my school account, which I use to create my citations, there will be no APA references until my password is, once again, reset and I am able to access school resources. Thank you for your patience.


    World Population Growth Throughout Human History


    Mathematical Function of Logistic Growth

    Logistic Growth in Biology

    Carrying Capacities

    Overshoot in Biology

    Die-Off in Biology

    Dynamics of Predation and Population Patterns

    R- and K-Selection