Background

Thomas Malthus

The first significant contribution to the theory of population ecology was that of Thomas Malthus, an English clergyman, who in 1798 published his Essay on the Principle of Population. Malthus introduced the concept that at some point in time an expanding population must exceed supply of prerequisite natural resources, i.e., population increases exponentially resulting in increasing competition for means of subsistence, food, shelter, etc. This concept has been termed the "Struggle for Existence".

Charles Darwin

Malthus's theories profoundly influenced Charles Darwin 1859, On the Origin of Species, e.g., the concept of "Survival of the Fittest". Mortality of this type can be termed "facultative mortality" (as opposed to catastrophic mortality, e.g., weather, insecticides).

Harry Smith, pioneering biological control worker with the University of California (1935), proposed the equivalent and now accepted terms density-dependent and density-independent. Density-dependent mortality factors are those that are facultative in effect, density-independent mortality factors are those that are catastrophic in effect.

A density-dependent mortality factor is one that causes a varying degree of mortality in subject population,and that the degree of mortality caused in a function (i.e., related) to the density of the subject (affected) population (density-geared, feedback regulation, self-regulating or self-limiting) may and typically involves a lag effect., e.g., most biological control agents.

Fig. 1.  Cycles in the population dynamics of the snowshoe hare and its predator the Canadian lynx (redrawn from MacLulich 1937). Note that percent mortality is an elusive measure, it may, or may not, be useful since mortality varies with environment and time.

Fig. 2A. Logistic growth (Nt = Nt-1 + Rm(1-Nt-1/K)Nt-1).  N = equal population density at a given time (t). This so-called "logistic equation" was first proposed by the mathematician Verhulst (1839).  In ecology texts this equation is more often written  as DN/dt = rm(1-N/K)N, where D is density at any given time (t).  K = carrying capacity of environment.  Rm = loge(Rm + 1).

Royal N. Chapman, Univ. Minnesota, in the 1930s proposed the concept of a balance between biotic potential and environmental resistance.  Chapman`s model was a mathematical representation of the Malthusian concept, illustrated here by the logistic growth of a laboratory population of yeast cells. Population growth trajectory (N₁ = N₀ + (Rm -sN₀)N₀ ); Rm = maximum rate of increase, here = 1, s = interaction coefficient, here = 0,0001, and carrying capacity of environment = 1000.

Fig. 2B. Population growth (N₁ = N₀ + RN₀), where N₁ = 10 and R=0.5 (blue), R=0 (black) , and R=-0.5 (red).

Human populations represent another example of exponential growth. Magnitude of the problems posed by human population growth can be seen from the fact that it took more than 1 million years for the human population to first reach 200,000 (the current daily rate of increase (See: US Census, Historical Estimates of World Population). The human population is estimated to have first reached 1 billion persons in 1830, and 2 billion in 1930, a doubling time of 100 years. In 1960, thirty years later, the population edged past 3 billion, and a mere 15 years later, 4 billion. In 1986, we exceeded 5 billion for the first time. Despite a slowing of the growth rate, it is expected that the human population will exceed 6 billion in early 1999 (see: Univ. North Carolina, Chapel Hill's World Population Counter) . To feed this population, only as well as we presently do, it will be necessary to increase food production 20% over the next 10-15 years.

Table 1. UN estimates of historical human populations and projections for 21st century
(UNDP Press Release, 10/28/1998)

1804
1927
1960
1974
1987
1999
2013
2028
2054
2100

1 billion
2 billion
3 billion
4 billion
5 billion
6 billion
7 billion
8 billion
9 billion
(range of estimates: 10.4-17 billion)

Exercise:  Compute annual population growth rates for each of the time perioids above:

Equation for population growth model is X=X₀e^rt where the original population is X₀ mathematical constant: natural log (e = 2.718), represents Malthusian parameter. Population will increase in size to X, over time (t), if rate of increase (r) is positive. If population not growing, i.e., r = 0.0, then rt = 0.0, and e^0.0= l.0, X = X₀. When r has a value greater than 0 population will increase, e.g., rt = 0.693 (e^0.693 = 2.0), population will double in time t (X = 2X₀). The doubling time of this population will be t = 0.693/r. If the population is decreasing, r is a negative value. The function of X = X₀e^rt can be made a straight line by the natural log transformation, i.e., lnX = lnX₀ + rt (the equation for a straight line). Characteristics of this line are: an intercept at lnX₀ and a slope of r. Linear regression analysis can be applied (since this is a straight line), with the independent variable being time. The linear equation thus describes the rate of population growth (+) or decline (-). Algebraic rearrangement of this equation permits one to solve for the rate of population increase.

In the 1920s, A. J. Lotka (1925) and V. Volterra (1926) devised mathematical models representing host/prey interaction. This was the first attempt to mathematically represent a population model as achieving a cyclic balance in mean mean (characteristic) population density, i.e., to attain a dynamic equilibrium. The Lotka-Volterra curve assumes that prey destruction is a function not only of natural enemy numbers, but also of prey density, i.e., related to the chance of encounter. Populations of prey and predator were predicted to flucuate in a regular manner (Volterra termed this "the law of periodic cycle"). Lotka-Volterra model is an oversimplification of reality, as these curves are derived from infinitesimal calculus when in nature association is seldom continuous over time (life cycles being finite).

Fig 4A (above). Steady-state population model (N₁ = N₀ + Rm(1 -sN₀ /K)N₀, where Rm = 2, K = 1000, and intitial displacement from equilibrium x = -10.

Fig. 4B.  Steady-state population model
(N₁ = N₀ + Rm(1 -sN₀ /K)N₀ ), when K = 1000, Rm = 3  and initial displacement from equilibrium x = -10.

Fig. 4C (bottom). Steady-state population model
(N₁ = N₀ + Rm(1 -sN₀ /K)N₀ ) , when K = 1000, Rm = 1.5 and initial displacement from equilibrium x = -10.

A. J. Nicholson (Australian entomologist) was a leading proponent of concept of density-dependent mortality factors. He maintained that density-dependent mortality played the key role in regulating prey populations. This is the essence of the so-called "Balance of Nature" theory. This theory implied a static balance about a mean (characteristic) equilibrium density with reciprocal (feedback) oscillations in density about these means.

Nicholson and V. A. Bailey (1935) proposed a population model that incorporated a "lag effect". This is particularly appropriate to parasitoids were population effects of attack (oviposition) may not be evident until the parasitoid has completed its immature development and emerges as a adult (killing the host).

Leading proponents of this view of population dynamics included the early California biological control workers. The theory and practice of biological control can be said to revolve about this assumption.

However, a contrary view of the nature of population regulating mortality factors was argued by others, especially W. R. Thompson (Dominion Parasite Laboratory). The Canadian workers held that assumptions of Nicholson and like thinkers were unrealistic, and did not occur in nature, i.e., that the regulating role of a so- called density-dependent mortality factors was largely myth. These workers argued that it was unnecessary to postulate such a mechanism of population regulation. They observed that the environment never remains continually favorable or unfavorable for any species. If it did so that population would inevitably become either infinite or decline to extinction. They maintained it was more accurate to say that populations were (in reality) always in a state of "dynamic equilibrium" with their environment.

H. G. Andrewartha and L. C. Birch were leading proponents of the concept that populations could be, and often are, regulated by abiotic factors.

These arguments became heated in the mid-1950s and early 1960s. Much of the confusion and controversy regarding mechanisms of population regulation stemmed from an inadequate data base, but an even more confining limitation was the essential impracticality of making the extremely complex calculations required to manipulate such data. Accordingly rapid advances in theoretical ecology, especially in the area of population ecology, only occurred with the introduction of high speed computers and the refinement of statistical theory.

Much of early population theory was developed deductively from controlled laboratory experiments or fragmented field observations. This resulted in a tendency to oversimplification and the development of models often not reflecting biological reality.

One of the most useful starting points for a population ecologist is the development of a life table. A life table is a schedule of mortality for each cohort (age group) of individuals in the population. The methods were developed originally for actuarial or demographic studies. Multifactor studies which incorporated the life table technique were once largely the province of forest entomologists, but are now widely used in agriculture.

The final test of any population model is its usefulness to predict (generation to generation) changes in abundance or to explain why changes occur at particular population densities.

Consequences of recent advances in understanding of population dynamics has lead to almost universal (among ecologists) acceptance of the proposition that population growth is geared to population density.

Differences in the relative importance of density-dependent and density-independent mortality factors varies in different environments, e.g., the role of biotic components tends to be greater in more stable (benign) environments.

Competitive processes/cooperative processes: we often think of interactions between individuals even within species as only negative, but interactions can be positive (even between species), e.g., defense against predators, genetic diversity (concept of minimum density), mate finding (sustainable population), early mortality may favor subsequent survival.

Interactions between species can be very complex even when only 2 species are considered (through impact on environment of other species). Each species can affect environment of other positively (+), negatively (-), or have no effect (0). Major categories include: mutualism (++), commensualism (+0), predator/prey (+-), competition (- -), and amensalism (rare) (-0).

Insects and plants of two types: 1) good colonizers, e.g., weed species, with high reproductive potential (capacity), adaptable, invaders, readily dispersed, "r-strategists"; 2) good competitors, high survival, tend to stable population (K = equilibrium), exploit stable environments, win out in competition, "K-strategists" (R. H. MacArthur and E. O. Wilson 1967).

Most crop pests are r-strategists, e.g., aphids and phytophagous insects in general. Natural enemies, i.e., parasites and predators, are mostly K strategists. This is said to be one reason for the high failure rate associated with introductions of exotic natural enemies.

Most crop plants are early succession plants, i.e., they are weedy species and accordingly they also are r-strategists. The r-selected species are particularly suited to exploiting the ecological patchiness and instability of the agroecosystem.

Fig. 5A + 5B. Island Biogeography: (MacArthur and Wilson 1967)

Equilibrium models for near and distant islands. Equilibrium (in number of species present) occurs where curves of rates of immigration and rates of extinction intersect. I is the initial rate of immigration and P is the total number in the species pool on the mainland.

The parallel between agroecosystems and island defaunation is obvious. Many factors affect the various population processes: number of potential invaders, distance of source of invaders, conditions for invasion and settling, attractiveness (favorability) of crop.

Effects of various regulating factors on the population dynamics of the host tend to differ: predators, harmonic (cyclic), parasitoids, intrusive, pathogens, disruptive, food, catastrophic (W. J. Turnock and J. A. Muldrew 1971).

Numerical response is the key to the success of entomophagous insects; occurs in the following ways:
1. Concentration, not different from sigmoid curve of functional response.
2. Immediate numerical response, increased survival
3. Delayed numerical response, increased reproduction.

Fig. 6 A + 6B.  Carl B. Huffaker et al. 1968 graphically presented four postulated types of functional (behavioral) responses of predators including entomophagous insects to prey density. (A) number killed, (B) percent killed. After Huffaker et al. 1968.

Nicholson curve: number of attacks determined by searching capacity (i.e., density-dependent). C. S. Holling (1965) disk curve: characteristic of invertebrate predators. Sigmoid curve: characteristic of vertebrate predators. Thompson curve: number of attacks limited by capacity for consumption or production of eggs, but not by searching capacity or host density.

Experimental data suggests that Holling's disc curve is characteristic of the predatory behavior of most entomophagous insects, i.e., the number of prey attacked increases but at a proportionately slower rate as prey density increases.

Interactions between species, especially between predators and prey are of great theoretical and practical interest to ecologists.

Environmental feedbacks: Populations degrade their environment, utilizing its resources. If the environment is highly favorable, resources are not limiting, i.e., the resources are replaced or recover faster than they are utilized. This favors rapid (exponential) growth of the population. As the rate of utilization of resources approaches the rate of replacement, the rate of population increase slows. Finally, population overshots carrying capacity of environment (resources depleted faster than can be renewed). This results in severe competition and the population collapses. When the population drops below the replacement value the environment recovers.

Fig. 7 (above). A reproduction plane divided into zones of population growth (R > 0) and decline (R < 0) by the equilibrium diagonal line (R = 0). The density of the population at equilibrium (K) changes in direction to the favorability of the environment (F). The equilibrum line is further divided into 3 sections with different stability properties: 1) a lower section where sK <1 providing asymptomic stability, a midsection where 1 < sK < 2 providing dampened stability, and an upper section sK>2  which is unstable. 

Fig. 8A (above).  A population trajectory on its reproduction plane showing growth over three time increments (N₀ to N₃) in a consistent envrionment and also following environmental degradation (broken line) at the end of the second time period.   The critical density (Nc, here shown as red line) is where the population utilizes resources at the same rate that they are renewed (replaced), for a given degree of environmental favorability.

Fig. 8B (above).  A population trajectory on its reproduction plane showing growth over three time increments (N₀ to N₃) in a consistent environment and also following environmental degradation (broken line) at the end of the second time period.  

Fig. 9A-C. Predator/prey interactions (figures redrawn from Berryman, 1981)

Fig. 9A. Reproduction plane for a prey species (A), where K_a is the carrying capacity in the absence of predation, W_b is the marginal cost of predation, P_b is the predator density which drives the prey to extinction.

Fig 9B.  Reproduction plane for a predatorspecies (B), where P_a is the minimal prey density needed to sustain a predator population, and Q_a is the marginal benefit of the prey.

Fig 9C. Reproduction plane for a prey species (A) and superimposed with a particular dynamic tragectory. An infinite number of curves could be drawn and the relationships need not be straight lines. The stability of the relationship would depend upon the characteristics of the curves.

Interactions (reproduction planes) between predator and prey. There must be some prey to sustain the predator. Pb is the predator density which drives prey to extinction. Ka is the carrying capacity in the absence of the predator. Wb is the marginal cost of a given level of predation. Pa is the minimum prey density required to sustain the predator population (and avoid extinction). Qa is the marginal benefit of prey. The superimposed reproduction planes show equilibrium lines for predator, Eb, and prey, Ea, and a particular dynamic trajectory.

We can draw an infinite variety of these curves, relationships not necessarily straight line, stability will depend on characteristic of curves (assuming these represent real life situation).

Community stability: A central tenet of classical ecology is that complex communities tend to be more stable (largely based on observation) theory recently challenged, argument that simple systems may be less subject to external disturbances. What has emerged is that:

1. Competitive interactions between species lead to instability unless dominated by negative feedbacks within self-loops, i.e.,  one species setting in motion cyclic oscillations in numbers of another.

2. The number of competing species increase the competitive interactions must be proportionately weaker or instability will result.

3. Interactions between tropic levels (predator/prey) tend to stabilize populations (community).

4. community stability is frequently interrupted by severe environmental disruptions, leading to a series of successional communities gradually evolving to climax associations. Frequent or continual disruptions may lead to persistent nonclimax community. Herbivores play an important role in plant succession by tending to harvest unthrifty members of a plant community, e.g., overmature trees.  They also recycle nutrients and increase productivity and vigor of community, - mutualistic?

Population dynamics/epidemiology theory is a vast and formidable subject. However, the synoptic model developed by T. R. E. Southwood (1975 and subsequent papers) is most instructive, and summarizes much of the theoretical concepts and empirical bases for contemporary model building.

The model has 3 salient features:

1. Natality (birthrate) is low at low population densities because of problems associated with low densities, e.g., finding mates, increases as population increases, peaks and finally declines as intraspecific competition increases, e.g., competition for food.

2. Predation increases with increasing host (herbivore) density, then declines as host populations overwhelm (and escape) their regulating influence. This occurs because the characteristic pattern is for overall predator response to be sigmoid, based on a) functional response of the individual predator and b) numerical response of the population.

3. Intraspecific competition increases with increased population (prey) density. This produces additional mortality as natality shows a down-turn. Other density-dependent mortality and stress also comes into play producing a marked increase in combined mortality.

The net result of these interacting factors is that changes in the population are subjected to dramatic changes in cumulative mortality rates. These are illustrated in the following figures.

Fig. 10.  Generalized relationship between population density of a herbivore species, natality, mortality caused by enemies (with a peak at moderatly low low population density) and intraspecific competition (with a peak at relatively high population density) (after Southwood 1975).

Fig. 11. Generalized relationship between herbivore population density in one generation and herbivore population density in the next generation. The line at 45 degrees is the line of no growth, i.e., population density stable from one generation to the next. Points above the line indicate population growth, those below the line population decline. X = extinction point, S = stable equilibrium point, R = point of release from natural enemies, O = oscillations equilibrium point, and C = crash point.

Fig. 12. Mortality of gypsy moth larvae caused by various natural mortality agents as the moth population goes from very low to very high densities, showing that mortality from disease and starvation becomes extremely severe in the dense populations. Note that vertebrate predators cause their highest percentage mortality when the moth population density is low.

Fig. 13 (above). Synoptic model of population growth (after Southwood and Comins 1976, J. Anim. Ecol. 45: 949-965).  The synoptic model of Southwood demonstrates the link between habitat stability (natural ecosystems evolving toward a K-selected type, agroecosystems representing an r-selected type) and relative favorability of each for pests and natural control agents. Pests having a relative advantage in r-selected habitats, while natural enemies tend to dominance in more stable ecosystems.