Model exponential and logistic population growth, determine carrying capacity from data, compare ecosystems in equilibrium and disequilibrium, and analyze how invasive species, predator-prey changes and human actions shift them

Learning goal: Compare ecosystems in equilibrium and in disequilibrium and give real examples of each, including Ohio animals and plants; describe the characteristics of exponential and logistic growth and use the logistic model to explain carrying capacity as the equilibrium population size at which births equal deaths; analyze population data to identify growth phases, cycles and carrying capacities; predict how predator and prey cycles change when either population changes and how humans alter them; and investigate an invasive species in Ohio, analyze the factors behind its success and its ecological and economic impacts, and devise a plan to lessen them or to restore an altered ecosystem.

Equilibrium and disequilibrium. An ecosystem in equilibrium is not static; its populations fluctuate around fairly steady values as births balance deaths and as predators and prey, or plants and grazers, respond to one another. Like many complex systems, it returns toward its state after a small disturbance. Ecosystems change as geological and biological conditions vary through natural and human causes, and human activity can increase the rate of the fluctuations. An ecosystem in disequilibrium is one pushed beyond the range it returns from: a lake where nutrient overload drives algal blooms that deplete oxygen and kill fish, a forest where an invasive insect kills every tree of one genus, a river cut off from its floodplain by channelization. Disequilibrium shows up in the data as carrying capacities that shift or collapse for many species at once. Ohio examples of equilibrium include a mature beech-maple forest whose canopy trees are replaced by their own seedlings; examples of disequilibrium include the forests of ash trees killed by the emerald ash borer since the early 2000s and western Lake Erie during summer blooms of toxic blue-green algae fed by phosphorus from farmland, which in August 2014 led to a two-day ban, beginning 2 August, on drinking or cooking with Toledo's tap water for more than 400,000 residents (NOAA).

Exponential growth. When resources are unlimited, a population grows by a constant fraction of itself per unit time: growth rate = rN, where N is the population size and r is the per-individual growth rate, births minus deaths per individual. The result is a J-shaped curve that rises ever faster; a population doubling every generation goes 50, 100, 200, 400, 800. Exponential growth is real but temporary: bacteria in fresh broth, a pest arriving in a new region, a species introduced with no predators. It cannot continue, because resources are finite.

Logistic growth and carrying capacity. When resources limit growth, the population approaches a ceiling. The carrying capacity, K, is the population equilibrium size at which births equal deaths, so the growth rate is zero. The simplest logistic model writes growth rate = rN(K − N)/K. The factor (K − N)/K is the fraction of the carrying capacity still unused: near 1 when the population is small, so growth is nearly exponential; near 0 as N approaches K, so growth stops; negative if N exceeds K, so the population falls. Worked in the prose with r = 0.5 per year and K = 1,000: at N = 100 the growth rate is 0.5 × 100 × 0.9 = 45 per year; at N = 500 it is 0.5 × 500 × 0.5 = 125 per year, the maximum, reached at half of K; at N = 900 it is 0.5 × 900 × 0.1 = 45 per year; at N = 1,000 it is 0; at N = 1,200 it is 0.5 × 1,200 × (−0.2) = −120, a decline. The population size over time traces an S-shaped curve: slow start, fastest growth in the middle, leveling at K. The competition that limits growth is both within the species (intraspecific) and between species (interspecific), for food, space, water and light. Features to label on a real growth curve: the lag, the phase of fast growth, the slowing, the plateau at carrying capacity, and, if present, an overshoot followed by a crash, the boom and bust pattern that appears when a population outruns its resources before the limits act, as reindeer introduced to islands without predators have done.

Reading population data. Given counts over time, a student can determine carrying capacity as the level around which the population settles, identify growth phases from the slope, and look for cycles. Predator and prey populations often cycle: hares increase, lynx that feed on them increase after a lag, hare numbers fall under heavier predation, then lynx decline for lack of food, and hares recover. Moose and wolves on an island in Lake Superior have been counted for decades and show the same coupling with added complications from weather, disease and inbreeding among the wolves. Predictions follow from the coupling: fewer predators lets prey rise until food, not predation, limits them, often followed by a crash; fewer prey starves predators. Humans alter these relationships by hunting large predators such as wolves, which released deer and elk populations across much of North America, by hunting large herbivores such as bison nearly to extinction in the 1800s, and by introducing a predator, as when Nile perch released into Lake Victoria contributed to the disappearance of perhaps two hundred or more native cichlid fish species and altered the lake's food web.

Invasive species in Ohio. An invasive species is a non-native species whose spread harms the ecosystem, the economy or human health. Ohio has many: zebra and quagga mussels in Lake Erie, which filter out plankton and clog water intakes; the emerald ash borer, whose larvae kill ash trees; purple loosestrife, which crowds out wetland plants; sea lamprey, which parasitizes lake fish; bush honeysuckle, which shades out forest wildflowers; round goby, which eats fish eggs; and bighead and silver carp, which threaten to enter Lake Erie from the Mississippi basin. Characteristics that make an invader successful: high reproductive rate, generalist diet or tolerance, few or no predators or diseases in the new habitat, ability to thrive in disturbed places, and, often, a way of escaping the controls that hold native species in check. Introductions come through ballast water, ornamental planting, released pets and bait, and hitchhiking on goods. Impacts are ecological, such as loss of native species and changed food webs, and economic, such as water-intake maintenance, dead-tree removal and lost fisheries. A plan to lessen impacts combines prevention (inspection, ballast rules), early detection, physical removal, biological control with tested natural enemies, and restoration of the natives, with monitoring to check whether the plan works. Stakeholders, from anglers to municipal water managers to landowners, need the predicted changes and the basis for the prediction.

Restoring equilibrium after human alteration. Ecosystems altered by dams, channelization, urbanization or nutrient overload can sometimes be moved back toward equilibrium. Removing a low-head dam reconnects a river, restores fish passage and sediment flow, and can be evaluated by fish counts above and below. Reducing phosphorus runoff by planting buffer strips, changing fertilizer timing and restoring wetlands reduces algal blooms, and progress is measured by phosphorus concentration and bloom extent. Evaluating a management plan for a recovered species asks whether monitoring, triggers and habitat protection are in place. Communicating to stakeholders means stating the goal, the action, the measurable indicator, the cost and the expected time to results.

Misconceptions. "Carrying capacity is a fixed number for a species" is wrong; K depends on the habitat and changes when the habitat changes. "Equilibrium means nothing changes" ignores the normal fluctuations. "Removing predators helps prey" is often true only briefly; without predation the prey can outrun its food and crash, unless other predators, hunting, disease or food limits hold it in check. "Invasive means non-native" is too broad; many non-native species cause no harm, and the term is reserved for those that spread and damage.

Investigation you can do. Grow duckweed, a tiny floating plant, in four jars of 500 mL pond or aged tap water on a windowsill, starting each with 10 fronds. Add no fertilizer to the first, and 1, 2 and 4 drops of liquid houseplant fertilizer to the others (the fertilizer supplies phosphorus and nitrogen). Count fronds every two days for three weeks and plot the counts. A result in which each jar grows fast at first and then levels off at a count that rises with fertilizer supports the claim that nutrient supply sets the carrying capacity; a jar whose count later falls, with cloudy water, shows disequilibrium as algae or decay take over. What the model cannot show: a jar has no predators, no immigration or emigration and no seasons, so it cannot show cycles or the effects of removing a species, and the carrying capacity measured is for a jar, not a pond.

Worked logistic growth rates: With r = 0.5 per year and K = 1,000, compute growth rate = rN(K − N)/K. N = 100: 0.5 × 100 × 900/1,000 = 45. N = 250: 0.5 × 250 × 0.75 = 93.75. N = 500: 0.5 × 500 × 0.5 = 125. N = 750: 93.75. N = 900: 45. N = 1,000: 0. N = 1,200: 0.5 × 1,200 × (−0.2) = −120. The rate peaks at half the carrying capacity and is symmetric about it, which is why the S-curve is steepest in the middle; above K the population declines toward K.

Worked reading of a growth curve: Hypothetical population counts over 14 years, starting at 50 with r = 0.5 and K = 1,000, run 50, 80, 125, 191, 280, 391, 514, 635, 742, 826, 887, 928, 955, 972, 983. Phases: years 0 to 3 nearly exponential (each year's count is close to the previous times 1.6); years 4 to 8 the fastest absolute growth, about 110 to 120 per year around N = 500; years 9 to 14 slowing toward a plateau. Carrying capacity is read as about 1,000, the value the curve approaches. Compare exponential growth with the same r and no limit: 50, 82, 136, 224, 369, 609, 1,004 by year 6, then 1,656 and on without end; the two curves agree closely for the first two years and separate as (K − N)/K falls.

Worked predator-prey prediction: A hypothetical island has 2,000 hares and 40 lynx that cycle with a period of about ten years. Prediction if trappers remove most lynx: hares rise well above their usual peak because predation falls, then overgraze their food and crash from starvation, possibly below their usual trough; without hares the few remaining lynx starve. Prediction if disease halves the hares: lynx decline within a year or two for lack of food; with fewer lynx, hares recover, and the cycle resumes, possibly with its timing shifted. Human hunting of the predator therefore does not stabilize the prey; it replaces a predator-driven cycle with a food-driven boom and bust.

Worked invasive species plan: Emerald ash borer in a hypothetical Ohio city park with 300 ash trees. Factors in its success: larvae feed under the bark where predators and sprays rarely reach, adults fly and are moved in firewood, ash trees here evolved without it and have little resistance, and no native predator controls it. Impacts: nearly all untreated ash die within a few years, removing canopy, altering the forest floor and costing removal fees; economic loss to timber and to property owners. Plan: inject insecticide to protect a set of high-value trees; remove and chip dead trees to stop firewood spread; plant a diverse mix of native species to replace lost canopy; release the tiny parasitic wasps that regulators have approved as biological control; survey trees each year and report the surviving fraction. The indicator of success is the survival of treated trees and the canopy cover regained after ten years.

Supported record: Draw the S-shaped logistic curve for r = 0.5 and K = 1,000 from the worked values and label the lag phase, the phase of fastest growth, the leveling phase and the carrying capacity; on the same axes draw the exponential curve for the first six years and mark where the two separate. Under it, compute the growth rate at N = 200 and N = 800 and explain why they are equal.

Independent performance: Investigate one invasive species in Ohio and prepare a stakeholder briefing. Success criteria: (a) it names the species, its origin and how it arrived; (b) it identifies at least three factors that contribute to its success; (c) it describes ecological impacts and economic impacts separately with the reasoning behind each; (d) it predicts what will happen to at least one native population and to the ecosystem's equilibrium if nothing is done, using the concepts of carrying capacity and competition; (e) it proposes a management plan with at least three actions and a measurable indicator for judging whether the plan works; (f) it names the stakeholders affected and states, for each, what the plan asks of them and what they gain.

Extension: evaluate a real management plan. Select a species that has been removed from an endangered species list; Ohio's model curriculum names the Lake Erie watersnake, the river otter and the bobcat as examples. Find the current management or monitoring plan published by the responsible agency (the Ohio Department of Natural Resources or the U.S. Fish and Wildlife Service), record its title and date, and evaluate it with the questions from this lesson: what is monitored and how often, which habitat is protected, what result would trigger a change in policy, and how removal from the list is expected to affect the species and its ecosystem. Quote the plan's own figures with their dates rather than relying on memory.