Forest Faunal Systems

Chapter 8: Section 3

Understanding Populations
Population Dynamics

Chapter 8 is long. For loading and other efficiencies, it has been placed into 3 sections:

• Section 1 - Introduction, the Nature of Populations, and Structure
• Section 2 - Converge - A Practical Guide to Estimating Population Abundance
• You are now in Understanding Populations, Section 3 - Population Dynamics

You may select any section.

Dynamics

In the definition of wildlife management is the premise that the manager manipulates rates. The major rates are the gross or summary changes and so-called "intrinsic" rates, survival, mortality, natality, and migration. Rates express the dynamics of a system, its change over time.

We may think of a population in the following way. Female (the males long-since departed except in a few monogamous species, apparently even excluding humans) gives birth to young. The females are symbolized as b, the young as c. The departed males is a. The relevant population, N, in the instant of birth, designated t, is

N_t = a_t + b_t + c_t

At a prior time, just one unit of time ago, designated as (t-1) the relevant population is

N_t-1 = a_t + b_t

We temporarily ignore a, it having died, formed bachelor groups, dispersed from the area, or continued to fill the odd spaces not occupied by b and c, i.e.,

N_t = b_t + c_t

We work toward understanding and being able to estimate N_t+1.

For a relatively non-aggressive species (most of them), it is easy to imagine an average distance of foraging, d, and thus a radius of r where foraging is done, on average, in a circle (Fig. 8.15). "On average" is a key expression. The manager or the behaviorist with radio-carrying animals, will say that individuals do not forage in circular areas. True, but false notions of requisite precision can freeze a field in its tracks. Snowballs are not spheres, but it does not prevent children from calling them"balls" and having fun with them. The area used by animals is irregular, but in dealing with populations of animals it is sensible to deal with some expression of central tendency such as the median home range. See Section 9 above under population estimation.

The female occupies some area, assumed to be a median-sized hexagon since a circle either overlaps other circles when animals are crowded together to fill up every available space and utilize every available resource (Fig. 8.15, b) or else it is tangetial and "wastes" the interstitial spaces (Fig. 8.15, c). A circle has the greatest possible area-to-edge ratio of any figure (see CAP130 and CAP2007) and thus would appear to have the least edge to defend and the latest motion-energy costs for foraging. It seems reasonable as a theoretical singular maximum value for occupied area. When a population is involved (as appropriate in all wildlife management), then the overlaps and blanks (Fig. 8.15, b and c) arise. The optimum trade-off, the survival and energy-effective strategy, an alternative theoretical maximum is the median-sized regular hexagon (Fig. 8.15 in yellow).

The female occupies the area H. This may be best conceived as a frequency-of-use "mountain" a volume where frequency of use probably diminishes at the edges. I suspect, that there are mountains of use-frequency that can be depicted within these packing polygons and that they will better represent the behavior of animals than the smooth forms usually depicted.

Fig. 8.15.a. A female with young may be considered to occupy a circular area with center or "refuge" R. An alternative is to assume the occupied area is a hexagon H (yellow). b. If circular, the areas occupied must overlap (as in competition for resources, shown in green). c. If the areas never overlap, there is wasted space or resources (the blue area).

For the present, however, let us assume there are median-or-average-size hexagons that are used (occupied) by our creatures. The female holds one area at t-1 and at t and does so until c reaches a traveling age (as in bears) and they all set off together exploring, foraging, and obtaining diverse educational experiences (MacNamee 1985), or else they disperse as sub-adults.

If 4 young are produced (c=4), half of which are females, and births occur every year and there was no mortality and this situation stayed the same for 14 years, the results would be as in Table 8.8.

CAP148, CAP151, CAP152, and CAP9088 allow you to experiment with these numbers and to study a few assumptions. You may want to plot these results to observe the shape of the progression. The equation producing these results remains the same; there is no change in the process - nothing "biological" - only an apparent change in the numbers. The so-called"lag phase" in an introduced population usually explained by low densities, etc., is just a function of the numbers, not the mysteries of the animal-habitat interactions. The relationship used to produce Table 8.4 is:

N_t = b

N_t+1 = N_t + N_t jc

N_t+1 = (N_t )(1.0 + jc)

where the first population N_t is the adult-female-with-zygotes (b), j the proportion of females among the young, and c the number of young per female. N_t is the population from the previous iteration to which are added the newly-produced young. Note that we continue here to follow females only. (The males are merely the (1-j) "dregs." See Appendix 2.)

Table 8.8. Changes in a population in which half are female, 4 births occur each year to every female, and no mortality occurs.
Year	Adult Females	Female Young	Total Females
1	1	2	3
2	3	6	9
3	9	18	27
4	27	54	81
5	81	162	243
6	243	486	729
7	729	1458	2187
8	2187	4374	6561
9	6561	13122	19683
10	19683	39366	59049
11	59049	118098	177147
12	177147	354294	531441
13	531441	1062882	1594323
14	1594323	3188646	4782969
15	4782969	9565938	14348910

This equation producing the above table is not a bad fundamental model since the proportion of females produced varies (or will vary) under certain managerial controls and their survival can be studied, or the converse, mortality, can be imposed on males or females each year. CAP148 provides the ground for exploring these relations.

If a rate, R, were estimated, for example by observing animals at time t, then at t+1

R = (N_t+1- N_t) / N_t

If there were about 250 animals at one point and 283 later, then R = (283-250)/250 = 0.13 or about 13 percent. That relationship is merely the algebra of the situation. R is a "black box;" there is no information about inputs, processes, or feedback. With no money, time, or curiosity, such computations may be sufficient. It can be used by managers willing to assume that the future is going to be approximately like the past. Using the net rate of change, R, one typical short-term estimate will be from the exponential relationship:

N_t = N_o (1.0 + R)^T

where R is the rate and T is the years the population is observed. A population of a total, say, of 3300 animals will be 14,300 after 12 years (i.e., 14,300 = 3300 (1.0 + 0.13)12). Exponential equations can produce a "gasp reaction" for population analysts as well as money borrowers. They are good for that purpose and maybe to set some upper constraint or a concept of the limits of a system, but little more.

The equation is almost nonsense because (1) real or net survival rates are very low because mortality of young is usually very high, (2) few animals live very long (say more than 2-3 years), and (3) the assumption of constant rates is absurd. (A stochastic element needs to be included in such equations.) Nevertheless, the manager might use it to help explain the present condition, predict the future, or, in turning to Chapter 9, manipulate the rate in order to meet objectives.

The literature on population dynamics is confusing when discussing stability. A stable population can be:

1. A population with the same approximate abundance over a stated period;
2. A population with an average rate of change (r) in abundance of zero, either in N_t = N_o e^rt or N_t as above;
3. A population with a stable age distribution (one in which the proportion of individuals in each age class remains the same). Such populations can change in abundance. The chances of obtaining condition 2 or 3 above are small.

Margalef (1968) showed that rate, R, is a function of births (B), deaths (D), and migration (M) so we expand the previous equation to

N_t+1 = N_t ( 1.0 + (B-D ±M)^T

which can be assigned an initial population N_t and iterated for as many years as desired. This begins to open the door to the faunal system manager because not only can the variability in these three topics (B, D, and M) be explored (as in a computer simulation) but also direct managerial action can be planned to cause change in any one, or hopefully, all of them simultaneously. The variability demonstration in CAP148 can be impressive since at least once or a few times in sequence in hundreds of years (iterations in the computer) it is likely that high mortality, low births, and high emigration can result in a small population. If such a sequence of undesirable conditions occurs for several years in sequence "by chance" (these chance events, while independent, are linked in the population), a population can become extinct even when under superior management. Similarly, "dieoffs" or "outbreaks" can also occur at which time disease, damage, or pest conditions are evident.

The sigmoid curve, so popular in most textbooks, is probably irrelevant for forest fauna and many other animals. A pair of invaders of a new, otherwise unoccupied region (a solitary animal is not a population, by definition) occupy an area (or n-dimensional space). The remainder of the region is irrelevant to the animals (although the manager may perceive that only a small proportion of "the area" is occupied). The initial lag phase of the sigmoid is mathematical, denying field conditions and reproductive success of animals with unlimited resources. Animals per unit area may be small, but the manager's question must be: in fairness, do I use total area available or area within the feasible home range or relevant faunal space, (e.g., CAP630 or CAPW03) as the divisor? When animals within their collective reasonable home range area is used, the increase is linear up to the home range constraint. Exponential growth does not occur since there is annual mortality. All animals born or hatched do not become producers. All adults do not survive. The population increase is linear. At some point, not likely pre-determined but annually determined by available energy (and other resources) for each animal (see the variable t in Holling's home range equation), the increase cannot continue. It flattens (analogous to the role of K in the sigmoid growth equation). When two animals eat from the same "food trough, " and when the food runs out and is insufficient to sustain life, one will die (a probabilistic event), then the other. Alternatively both may die (the instant of death being the only real question). Whether the population declines by 1 or 2 is problematic. Other ambiguity results from inefficient spacing of home ranges (overlaps and gaps as described previously) and the surplus and insufficient areas of the boundaries of woodlots or other landscape "islands" where area exists but cannot or will not be used by the species progressively filling space over the years until it is all used up. "Used" is the key word, because animals, like people, find space and resources, while present, unavailable for many reasons. The ambiguity can be seen in the field as well as in computer runs. Populations are largely, I believe, a series of linear performances. We see them in toto; we strongly desire simplistic, parsimonious explanations. We readily convert two linear expressions of

N_t+1 = b N_t

subject to that when N_t+1 > R/n

then

N_t+1 = N_t

with variation (probably about 10 percent) in all parameters. The conversion (Fig. 8.16) is to

N_t+1 = c N_t^k

or

N_t+1 = T^v

Fig. 8.16. Two equations representing different states of a population with changing conditions may be approximated by an allometric equation (right) Parsimony may win over realism (left) and detract from the real changes occurring in populations.

where c is a modifier that may be used to adjust the expression of the average animal chosen (weight, proportion of females, median age, etc.) and k is the coefficient that "fits" the curve to actual or hypothesized relations. Similarly, T is the year since starting, the origin is zero, and v equivalent to k in the above equation. The value k (or v) can be "played with " by considering it to be k = (a birth rate - b death rate) but this takes the manager back toward particular parameters that must be estimated. Curve fitting is done to gain parsimony. What could be more elegant than to know k or v for a species in an area? Such hopes are fanciful. Armed with computer, the forest faunal system manager needs to work with the differentially variable parts of the population subsystem. Cross Current - Curve Fitting vs. Theory Building

Natality

Red fox kit.

Almost unconsciously, the phenomenon of rates emerges from simple discussions of an existing population and its change due to births. That rate is called natality. Natality is birth rate, usually by age class. It is preferred to be expressed as young per female of age class i, but often in field work, young per adult is all that can be obtained. Secondary data on the sex ratio of adults can be used to estimate the refined birth rate, m_i, where:

m_i = m' _i/S

Here, m'_i is the ratio of young to all adults (e.g., 2.1) and S is the proportion of females among the adults (e.g., 0.60). When age specific natality is not available, averages may be used with reasonable adjustments such as for unlikely reproduction in very young and very old animals.

The symbol m here expresses actual births at some relevant time or period such as "in feather" or "2 weeks after partuition." Simple "birth" is about as meaningless to the manager as "ovulations." The rate of change in total potential offspring from corporea lutea to "time of first suckle" is likely to be extremely great with more phenomena at work than can ever be addressed specifically within forests. Even estimating large mammal natality is difficult. Often fawn-to-adult ratios are used. A proportion of over 0.33 fawns is usually desirable for a stable or increasing population. For a population to replace itself, each animal must produce one offspring in its lifetime. Considering unequal sex ratios and long life, a criterion of a ratio of about 0.3 to 0.4 young-adult can be judged useful for evaluating stability. Where X is the critical proportion of young needed (the ratio) and 1000 animals must replace themselves, j the proportion of females (e.g., 0.65) in the natality per adult, and ecological life expectancy (E) is 4 years, then

1000 = 1000 the desired condition
1000 = (1000 x j) E
1000 = (1000 x 0.65) (4)
X = 1000/(1000) (0.65) (4)
X = 0.38.

CAP180 allows alternative numbers to be readily evaluated.

Mortality and Survival

Survival was discussed above. It is the rate of persistence, the average probability of an animal being alive after a specific period where 1.0 exists only in the instant of birth (or definition as a recruit to the population). Survival is (1.0 - mortality). It may be computed for a total population of young since the time of birth or as that within each age class. One way is cumulative, the other age-specific. Because hunting, pollutants, and human-related causes of death can be age specific, it is useful to model populations using age-specific mortality. Such mortality is exercised only on the animals moving into an age class (they having been reduced by various mortality factors in the previous class.) Average mortality can (or needs to) be used in many situations since clear age-related data are difficult to obtain. However, data from aged carcasses (weight, size, tooth wear, bone development, etc. (Larson and Taber 1980)) collected over a long period can often be helpful.

The mortality factors are hunting, crippling, poaching, predation, accidents, starvation, disease, parasitism, pollutants, drowning and many interactions (such as a broken bone leading to stress, parasitic increase, and starvation.) Animals die with these conditions but most often from suffocation (as from pneumonia) or from nerve damage or disfunction.

The system manager with objectives of increasing a population will typically, simultaneously attempt to reduce these mortality factor influences. The manager's most simplistic population model is

N = I (M* - D*)

or "initial population (I) with births minus deaths." It is very similar to Margalef's suggestion discussed above. Working with deaths, expanding the model to include all factors and their coefficients, and then developing projects to regulate them is not a bad strategy. It can constitute an important subsystem in a coordinated effort to gain control over a population (one dealing simultaneously with the objectives for N, I, M*, and D*).

Control

Gaining control over a system is the subject matter of cybernetics. In Chapter 4 and 5 the negative feedback model symbolic of desired faunal system control has been presented. It was:

Q_t+1 = Q* - (S-K) (Q_t - Q*)

Here we replace it with its isomorph, a population model:

N_t+1 = N* - (S-K) (N_t - N*)

N* is number of animals that is the objective. (It could be multi-definitional "carrying capacity" or equilibrium value). We need to deny again that it is not animals for which the faunal system manager works and to reassert that it is human benefits. The population is changed to an estimate of a resource by multiplying it by a non-linear expression of benefits. N_t is the current population and N_t+1 is the population in the next year. The manager seeks to gain some level of control, K, over the tendency of the system to remain stable (S usually is 1.0), to control the difference between the current population and the desired one, (N_t - N*), reducing it to zero. It may be useful to see K as the birth rates minus death rates. When approximately equal, K is approximately zero and the system remains stable. By changing natality or mortality, at whatever expense, the manager may gain control over the system. The control is symbolized as K.

Deciding on N*, another place for control, is difficult. An entire Chapter (4) was presented about that difficulty. Soon after that decision comes the computation of N_t. The simplicity of this fundamental population model has major advantages for teaching, mental organization, and preliminary problem analysis, but it does little to help the manager decide actions to take. Perhaps this is too critical an overview. Perhaps here is where the art of faunal management may be claimed. The equation is large, and becomes larger as N_t is computed. There is rampant equifinality. Costs are rarely known; estimates are more common than conclusions. While it is possible (and needed in the future) to create optimization models, objectives are essential for doing so. I now realize wildlife agency objectives are poorly formulated and not fixed. (I once saw an agency change its objectives in a computer model to "prove" that what it was currently doing was optimum!) The variance in the data is great. The solution will be insensitive to changes in most variables over a wide band. Nevertheless, an effort at understanding all of the major parts of the system needs to be made, an understanding of the complex system gained, and the important factors identified. The modeling effort can show where work is needed, where precision is important, where expenditures must be made. The effort can compute an optimum, a standard, and current population status can then be compared to that. A score will result, an expression of managerial excellence. Such scores have an alluring way about them; they beckon managers to the resource system high grounds.

As will be amply demonstrated (if not already), trying to comprehend the structure, dynamics, and relations of a population sufficiently for management is extremely difficult. I suggest starting with some very simple ideas. Often that is as far as the manager can go because of limited time, money, and energy. It may be as far as we can go because uncertainty in one part of the comprehensive population system may be so threatening to a user that work on the rest will seem irrational. Of course there are innumerable "political" blocks.

Suppose the population N can be crudely estimated. Rarely (if ever) do we start with zero as in a stocking or transplant program. The forest faunal system is typically composed of a large multi-life-group population (greater than 1000, i.e., P* = 7).

When to start the analysis is an important decision. Because productivity or the proportion of new animals by which a population increases is often of interest, a starting time of April or May is selected, at least just before young are born. Some models analyze post-hunting or trapping populations but mortality in the last parts of the year and pre-conception can be extreme.

The abundance is estimated using heuristic convergence discussed in section2 in this chapter. Lacking other criteria for selecting a set of estimation criteria, I suggest using the hexagonal home range approach. The total area of interest, the forest management unit, becomes the determinant of a first approximation of abundance. The population is logically composed of males, N_m; females, N_f; and young of unspecified sex ratio, N_y

N = N_m + N_f + N_y

Having decided to work with the female part of the population, a module may then be developed expressing the entire population in these terms:

N_t+1 = N_t (1.0 - F) + N_t F + N_t FM (1.0-m) + N_t FM m

where F is the proportion of females determined from field observations, N_t is the total population estimate, M is the average natality per young female, and m is the proportion of females among the young.

CAP9086 allows this 4-factor model to be explored. It begins to synthesize the results of using the CAPPER disk programs related to home range, heuristic conveyance, sex ratios, ratio significance, and natality estimates. It is a 1-year model, however. It simply adds the old males, old females, and young produced and produces the likely population for next year as if these were the only factors involved.

In order for this model to be run iteratively showing the total dynamics over many years, the results from one year must be fed back into the system for the next-year run. Revisions are needed. The young must be added to the adults. The adults must be separated into age classes. I suggest three classes because of costs, limited age criteria, limited data, and other factors. Hunting mortality must be allowed as well as poaching, crippling, and "others" losses. Limits must be set on how many animals may occupy the available faunal space, the volumes, within the forest.

Before making these complicated changes, let us "start at the end" (see Chapter 6), considering another simplistic model to provide an answer to a typical question of faunal system manager. What is the "yield?"

Suppose the population (N) can be crudely estimated (e.g., density (D) and hectares (H)). Also, usually the sex ratio can be estimated (CAP09 and CAP148) and changed to proportion of females (F). An estimate of the number of young per female (M) can be gotten from observations of fawn:doe and calf:cow ratios and from direct observations of placental scars, etc. Thus the relationship of annual production or yield (Y) is

Y = D H F M

A safety contingency can be included either for a catastrophic or unexpected, unexplained loss (C) and the proportion of females may be restricted so that all may be bred by the residual males. Thus Y becomes:

Y = D H F' M (1.0-C).

To estimate the proportion of a population that may be removed each year (H) and yet retain the same population as before the annual production, then

K = Y/(Y + DH)

This equation is deceptively simple. F may be considered in terms of productive females shortly before annual births begin. (This is not the usual time when observations of sex ratios are made.) Other adjustments are needed in births because the actual number of animals born that are still living at the time of the beginning of the birth period is the relevant number (e.g., the deer fawn population born but having experienced the high mortality of the first 6-9 months). For example, in a 3000 hectare area where density is 1 per 20 hectares, the population having 0.7 females, the natality 1.1, and a 5% safety contingency desired, then yield, Y, is:

Y = (1/20) (3000) (0.7) (1.1) (1.0 - 0.05)

Y = 110

The proportion of the population that may be taken safely is K and

K = 109.7 / (109.7 + 150)

K = 0.42

This K means that 42% (i.e., 0.42 x 100) of the population can be removed, safely, and next year, the same factors working, the population is likely to be approximately the same size. These relationships can be explored in CAP9087.

This set of computations is relatively straightforward. Recent theorists deride such estimators on the grounds that after harvests are made, the population density is reduced, more food per animal left alive in the forest may be available, thus reproduction per female increased, and the population may rebound more rapidly than estimated by the computations. Perhaps. That it probably occurs is a built-in managerial safety factor and explains why managers have been so successful, experienced so few "wipe-outs," and why area closures after an apparent over-harvest are so successful. The compensatory relations are discussed by Peek (1986).

We may utilize the simple relationship already explored to include the change in rate for the next year resulting from the change in density in a year.

N_t+1R = N_t ( (1.0 + r) x (1.0 + X (N_t/area)) )

The value of X symbolizes a coefficient which is an approximate linear function of density (N_t/area). When X is 1/density, then the model performs in its standard fashion. When the density is decreased, then X is large, symbolizing the proportionate increase in available resources per animal. When the population density is allowed to become large, X becomes minus thereby decreasing resources and influencing negatively birth and/or deaths. Here the relations are linear, a practical strategy for managers.

To obtain harvest estimates which are almost essential for managing game populations, data may be gotten from checking stations (samples), required reporting of all game takers (e.g., of big game), and questionnaires. The latter usually are sent in"waves" and responses from each wave are usually different. Techniques can be used to obtain cumulative weighted responses and to project responses for all people sent questions. Hunters, trappers, and faunal enthusiasts are notoriously cooperative in returning questionnaires. Telephone and other interviews and secondary questionnaires have been used to develop adjustments to correct for error or bias in first responses.

Achieving a large, stable harvest over a broad area (or state or region) may be desirable. One approach to solving this problem is to plot the logarithm of the harvest per square kilometer in each county or management area against the log of the harvest in the previous 2 years. See also the section on"Rates" under population estimation above. The latter is viewed as the "independent variable," the driving force. A graphical (Fig. 8.17) or computer solution for the inflection provides insight for a harvest recommendation. This is a "black box" approach. The environmental and other causative factors are not expressed except as they are included in the previous-year harvest which is viewed as a synthetic index to the many complex factors at work. More comprehensive models are needed (RAMAS (Exeter no date), POP II (Bartholow no date), and many others are now available) but sometimes simple answers, quickly gotten at low cost, are all that seem to be affordable. Appearances do deceive.

The general dynamics of a forest population may be inferred from harvest data. Where harvest, H, seems reasonably well correlated with the actual population, then a simple linear regression analysis will yield:

log (H_t+1) = a + R log (t+1)

and thus R is the rate of change, a is the intercept, and t is the year of observing each value of N_t. CAP110 can be used.

Fig. 8.17. Harvests per unit area (or otherwise adjusted and controlled) can be plotted. The vertical line is where harvests are the same as previous years. Locating the approximate intercept of the plotted points or their curve suggests a desirable future harvest if a large, stable harvest is desired. Chapter 11 suggests techniques for achieving such a harvest.

Based on available food, the estimated population, and potential yield, past harvests, safety considerations (for hunters and the manager), and other factors, it might be estimated that a harvest, H, of 1200 animals is needed from an area. As discussed in Chapter 9, there are many ways to regulate hunts to gain a particular harvest and its associated primary and secondary benefits. Here, using hunting permits is discussed for it is specific. Realize that a group of hunters influenced by many regulations may be equivalent to one hunter with a permit.

The permits to issue, P, is

P = (H / (S (1.0 + U + C))) / p

Where H is the desired harvest of animals, S is the likely hunter success rate, U the unreported or illegal take, C the crippling loss, and p the proportion of permits issued and picked up by hunters. Thus, in one case:

P = (1200 / (0.22 (1.0 + 0.3 + 0.11)) / 0.94

and

p = (1200 / 0.3102) / 0.94 = 4115

Thus, without poaching or crippling losses, and counting non-show hunters, 5802 permits could have been issued, whereas only 71% of this was actually issued to take the desired harvest. As usual, all five components of the equation can and do vary annually due to unexplained reasons, weather, enforcement effort, education, and the characteristic of the random draw of individuals (CAP112) for permits. Issuing permits can be explored in CAP20 and CAP5012.

To gain control over the population system, a manager will want to change N_t. One way to envision the managerial quest is a change of D, a system performance score, toward a value of 100, where

D = 100 - ((|N_t - N*| / N*) x 100)

(The vertical bars symbolize absolute value or always assigned a positive value.) Achieving the change is rarely likely to be done quickly because of the many diverse areas and populations with which forest faunal system managers work. To do so requires mastering the dynamics of N over time.

It seems reasonable that many of these concepts need to be brought together into a simple system. CAP9088 has been developed to do this. It allows a manager to look ahead for 5 years. The following suggests its use and interpretation. Other models simulate change over 100 years but the hyper-skeptical comment that is usually received from the field is that "we cannot see very far into the future; conditions are too uncertain!" Yes, but the future population is very much a function of the animals now present, their ages, sexes, and health. The 5-year period was selected because that is the short view now in practice. It can suggest what will happen in the manager's near-view mirror because some population parameters will change in the near-future if the manager takes certain direct action.

Only three life stages or age classes are used. See Fig. 8.18. Sample sizes are too small, ages too imprecise, and sampling too erratic over time (within a year and over several years) for other analyses in the normal forest situation. The graph of age structure appears as in Fig. 8.19.

Six population units are dealt with, one for each of the 2 sexes and 3 age classes. A variability coefficient is included with each factor. This allows a user to specify the amount of variation allowed to be assigned at random. In computer runs, this allows a number to vary plus or minus a percentage, probably somewhat simulating the variability encountered in the forest. Only one variable, v, is shown (e.g., varying a factor plus or minus 5% (0.05) in a random, uniformly distributed way each year¹.

Variation is available in the program but users are encouraged to use the system first with zero variation, then explore consequences of 10 to 20 percent annual differences. These are plus or minus and affect almost all factors that are operative within the system (both in the forest as well as in the computer).

Fig. 8.18. The fundamental population model. Great complexity can be added to such models. Only minor expansions have been made in CAP9088 such as for sex and age differences. As an example, available high quality forage differences may be hypothesized and their effects studied by changing the number of young produced per female in each stage and by changing the survival rates.

Fig. 8.19. Age distributions in an age pyramid showing proportions in three age classes.

Users of CAP9088 and similar computer models will quickly discover that the entries that must be made are oppressively numerous. Changing one variable in a model is sufficient for most purposes; re-runs can be made holding all factors constant but increasing or decreasing the variability. Reprogramming to allow different ranges of variability for each factor is encouraged. The manager must not lose sight of: (1) Benefits are rarely linearly related to N_t, or (2) that N_t is often a species but can be computed with median values for a life group, for a guild or similarly functional group of forest fauna, or for a group of animals with indistinguishable potential human benefits (e.g., the fall warblers; the non-poisonous snakes of Ranger District Z).

The first estimate of the population used in CAP9088, the one needed to initiate the work, N_t, is

N_t = A (1.0 + v) /H (1.0 + v)

Where A is the management area and H the home range of a family group of 2. The manager may assume the area is fixed but there are usually uncertainties in the boundaries, changes in land use and ownership, and even in assigned areas of responsibility. Zard 15. Home range is variable; it can probably be influenced by managers. N_t may be assumed to be more than 20 animals.

The population can be estimated grossly. A hypothetical cohort of 1000 animals can be used (quickly gotten by entering 1000 acres, then entering 2 animals per acre in response to the questions). As a test case for demonstration, I shall use a 12,000 acre area and assume my creatures are using about 75 acres. This results in a population for study of about 320 animals.

From one perspective, N_t is shaped by traditional concepts of "carrying capacity," and is a computed upper population level, sometimes called K in the ecological literature. Herein it may be a constant, but I prefer (on the grounds of what happens in forests) that it change each year. CAP9088 allows users to play with estimates of the population, N_t, based on energy relations and individual or animal family group needs.

The population manager starts with area, estimates the area that is needed by pairs of animals (no asumptions about breeding behavior are needed), then divides to get total population. The population is seen as three life stages or age classes, young, sub-adult, and adult (Figs. 8.18 and 8.19). The proportions estimated for the three age classes are critical ... as is everything else in the model. The model has been reduced to the fewest number of profoundly influential factors so each estimate has a profound effect. We can use the proportions of 0.31 for the first class, 0.38 in the two-year long sub-adult class, and 0.31 in the adults as in Fig. 8.19 in an example, e.g., as for bears. Roughly 0.3, 0.4, and 0.3 are appropriate approximations but these are areas for study (for example, what difference does this difference in the accuracy of the proportions estimated really make?). The three classes are separated, based on sex ratios observed, into male and female, resulting in 6 groups. In a hunted population, even where the majority of people comprehend the problems of bucks-only hunting, there will be unbalanced sex ratios. For a deer population we might estimate for one area about 0.6 females. For some trapped species, the ratio will be unbalanced in the other direction. Births differ by age. After births, mortality is computed based on (1) reported mortality of young, and (2) linear estimates in the two other classes based on a proportion of the time in the class. The duration of the sub-adult class is generally known and thus entered. These years, plus 1 for the young, are subtracted from the estimated ecological longevity to get the duration of the adult class. Experts can estimate ecological longevity. I use 8 years for the demonstration. As for sub-adults, the adults survival (1-mortality) is determined based on the total in the class and the assumption of an equal proportion of the class being lost each year.

The counts are shown in CAP9088 as:

Stage	Males	Females
1	40	60
2	49	73
3	40	60
Totals	129	193	Grand Total 322

The time as a sub-adult for many U.S. faunal populations is only 1 year. This number is entered. "Sub-adult" is a grossly used phrase and typically means late-season young and animals slightly over 12-18 months. It implies sexual maturity but that is not intended here and probably not elsewhere. Breeding"fawns" are noted. Sexual maturity is very much a matter of health at birth and early nutrition and of time-since-parturition or hatching. Here we enter 2 years for a demonstration, implying that some young do breed but there is a high proportion that are still developing physically and behaviorally.

The proportion breeding is assigned, for demonstration, as 0.2. This number, for example, can be influenced by changing faunal space conditions, e.g., food quantity and quality.

All adult females in any population do not breed for many reasons, including physiological blockages related to lactation, health, and (some claim) to low densities presenting adequate timely opportunities for copulation. I estimate, for demonstration, 0.7.

Faunal space (Chapter 7) influences the health and other factors influencing average births per female. I estimate 1.9 for demonstration (e.g., 19 fawns for every 10 does seen).

"Primary natality" is the expression of number of young per female just before birth. Insecticides and other factors reduce this. There is high mortality in the first hour. In the model, I try to suggest that the young seen - say after one month - is the realistic number counted by most managers. These are full size fawns, elk calves, turkey broods, fledgling birds. In the demonstration, I entered 1.9 above, but I think the real number produced is at least 10% greater. I enter 10 in response to the question.

The proportion of young that is"female" easily varies due to chance (and probably nutrition) and so a proportion can be estimated. I only enter 0.5 in the demonstration.

The results:

YEAR 1

Age Class	Males	Females	Young
1	40	60	0
2	49	73	28
3	40	60	80
Totals	129	193	108

The ratio of young produced late in the year to the total population is 0.33, or 0.31 when considering only females produced by females before any mortality.

YEAR 2

Age Class	Males	Females	Total	Young
1	60	58	118	0
2	62	73	134	28
3	57	85	141	112
Totals	178	215	393	140

About 156 young were produced (140/0.9) before any mortality occurred.

YEAR 3

Age Class	Males	Females	Totals	Young
1	78	76	154	0
2	79	83	162	32
3	76	104	180	138
Totals	223	263	496	170

YEAR 4

Age Class	Males	Females	Total	Young
1	95	93	188	0
2	98	99	197	38
3	100	125	225	166
Totals	293	317	610	203

YEAR 5

Age Class	Males	Females	Total	Young
1	113	112	225	0
2	119	118	237	45
3	129	149	279	198
Totals	361	379	741	243

The interpretations are numerous, the delight and magic of the computer simulation. (Managers will typically be asking, "What if these conditions prevailed? What can I do to influence them as the system manager?")

The total population has increased from 393 to 741 in 5 years. The sex ratio, once 0.6 females, has shifted to 51% under a no-hunting assumption. The number of young produced per female has shifted from 0.65 to 0.64, but the total young is increasing rapidly. The production per animal has shifted from 0.36 to 0.32 but his has been caused by the change in the sex ratio, not some nutritional or community health change.

The population is in a run-away condition but it appears that in the next year (not shown), the 243 young, equally allocated to the male and female column will about match the present numbers of 113 and 112 and these bare replacements will slow the population. For the near future (5 years), increases seem reasonable.

If we use all of the same numbers but allow variation of 15% in our estimate, then the changes appear as follows:

YEAR 1

Age Class	Males	Females	Totals	Young
1	40	60	100	0
2	49	73	122	28
3	40	60	100	80
Totals	129	193	322	108

YEAR 2

Age Class	Males	Females	Totals	Young
1	60	54	118	0
2	62	73	134	28
3	55	82	136	108
Totals	176	212	388	136

YEAR 3

Age Class	Males	Females	Total	Young
1	76	74	150	0
2	78	82	160	31
3	72	97	169	130
Totals	226	253	479	161

YEAR 4

Age Class	Males	Females	Totals	Young
1	89	88	177	0
2	94	95	189	36
3	93	114	207	152
Totals	276	297	573	188

YEAR 5

Age Class	Males	Females	Totals	Young
1	105	103	208	0
2	112	111	223	42
3	116	133	250	177
Totals	333	347	680	219

There is little over 8% difference (less) in the total animals in the deterministic and stochastic run (580-741/741) but 9.4% difference in the young produced. Totals and proportions are re-computed and the production-survival system begins again.

Five years may seem a very limited period. Long term cycles and trends and extinction phenomena are interesting to compute, but the forest faunal manager who does not exercise some control in some part of the system to expect significant desired change may not be in touch with human expectation - either from the animals or from the public or client. Variation in these population systems (as users will discover), even small amounts, can produce results that throw into question whether any manager can control a wild animal population.

Regulation

Every life group is regulated. There are genetic limits to reproduction, limits to food consumption, and to metabolism. These are maximum values, characteristic of the animal, not the habitat. Many environments (I suspect most) have excess food available to animals. Hardly ever is there a report of 100% browse removal or animal-caused plant extermination. The animals can only move to, feed upon, masticate, and otherwise process so much food. The more sparse or the lower the food quality, the longer it takes and the more must be consumed because the quality decreases. The "best food" is consumed first (or in the sequence in which it occurs and is encountered in the forest so as to reduce the energy costs to gather it¹. "Serving line" optimization (from operations research) combined with ability of a map cell in a geographic information system tp "serve" an animal will likely provide insights into foraging theory and how managers may change conditions in map cells). There is excessive emphasis on the forage limitations to regulating animal populations. Equally or more important are the energy losses or drains, the conditions of the faunal spaces that prevent survival. There are other factors preventing populations from reaching theoretical maxima.

I prefer to study and discuss the functional dynamics of populations since "population regulation" questions can be leading. What regulates populations? Is the connotation of "regulation" negative? Both questions suggest the teleological grounds of existence and a singular "if you look hard enough, you will see the answer." The good answer to the bad question about regulation is "almost everything; anything in a unique situation; it will be different tomorrow." Neither very parsimonious nor very satisfying, the answer must be population specific, tentative, and expected to vary. There are theories about population regulation, probably efforts to express system sensitivity. The categories of regulatory theories are:

1. Environmental factors, primarily food and space but also abiotic factors such as growing season and degree days. Solar and lunar forces, extra-Earth, are influential but whether they qualify as regulatory seems doubtful.
2. Interactions within populations that restrict or increase food available per capita, and that involve defended areas, crowding stress, and toxic substance levels. Other animals of the same life group are very much an"environmental factor." Genetic theory does not have to be invoked to see how individuals, experiencing noise (calls, etc.), fights, or courtship displays might be stressed and have lower survival than others, thus a population is"selected" by the whole environment.
3. Interactions with other populations, increasing energy costs of survival, predation and parasitism, and joint use of food and spaces. In a few cases, the interactions are positive such as animals using foods left after foraging by others and multi-species use of animal-created dense trails and runways.
4. Dispersal, but few people observe or report faunal immigration or invasions. (Comments are usually made about animals being"driven out.") There are exceptions, but that is one point of making this list and the grounds for a broad, multi-factored concept of functional dynamics.
5. Antropogenic, varying among hunting, setting fires, changing available spaces, changing quality of spaces (food species composition or secondary chemical levels), and disturbance levels and frequency.
6. Disease organisms, many of which are omnipresent, are considered by some to be as intrinsic to the population as the adrenal gland is to an individual and equally as functional. On demand, they may suppress a population, or themselves wane, and thereby allow a population increase. The disease can be viewed as another species in the system (see 3 above); whether internal or external makes little difference.
7. Lagged phenomena, though in a different class of regulatory concepts. They are both mathematical (predictable sine-wave-like distributions of occurrences) as well as indeterminate. The latter implies that the change is real, has a provable set of causes, but that the events are so far in the past that sufficient historical reconstruction is impossible, thus the cause, while it could have been known, remains unknown among a set of possible causative scenarios. This is quite different than the situation of "I don't know!"

Renewed interest is needed in the population itself, the foraging, energy budgeting and reproductive capabilities of individuals, as well as the spaces of animals. There are no simple answers to problems of the faunal resource. Neither "habitat" nor "population" approaches will suffice. Alone, they result in "simple-minded maximizing." The needs are for a systems approach, efforts to work with the whole system for realistic and lasting solutions.

Competition and Predation

One aspect of population dynamics or"regulation" is interaction with other animals (see No. 3 above). Competition is a willful or purposive effort to take food or space that is similarly needed or wanted by another animal and to prevent that animal from having it. I have only seen this behavior in the felids and canids. It probably occurs in mustelids but the point is that it is not an all-pervasive phenomenon. A deer eats grass, unmindful of whether a cow or elk is nearby and may or may not eat that same bunch of grass. I view a genuine competitive situation as two or more wolves pulling on the same piece of meat. There may be a winner or sharing (all getting an equal proportion), but there is energy expended as a result of the presence of another animal of the same or different species, not merely the energy of discovering or capturing food. Successful competition may result in extra gains (surplus), needs precisely met, or conversely, losses. I cannot conceive of mutual foraging in a resource pool as competitive. Trying to do so may lead the student off the track into difficult and unproductive studies.

Analyses of animals foraging in a productive environment result in a description of a food base that may be equivalent to the base in an environment that produces little food. When concentrating on one animal population, the effect of other feeders on it is best accounted in the environmental dimension, as some proportions, f, removed by other foragers. These foragers include the insects, and other grazing livestock. It is easy to add similar removal functions such as repellents and fencing. What limits a population is what food is available to it, not its potential or actual presence.

Competition, if it must be used as a concept, rarely occurs and is best viewed as another environmental factor (CAP102 and CAP9050). Animals are part of any other forest animal's space. Some environmental factors are beneficial, others harmful. The manager's task is to control populations and their requirements. Each animal can be allocated a proportion of available resources. Some of these are integer resources. They only occur in whole units or pairs. A fox does not take a half of a turkey. A turkey does not swallow a half of an acorn. Extra"range" available is of little import to a family group occupying a full home range.

Competition is an interesting idea. If known to occur and clearly demonstrated, it can be included in models to show how one population can be influenced by another (regulated?). Given the variabilities and uncertainties in the other truly profound dimensions of population dynamics, competition can probably be put on a back burner for later study or for development in a few critical areas of population manipulation.

Predation needs similar placement. Predation means foraging on animals. A predator is an animal that eats an animal in which people are interested and for which some person has an increaser-objective. Otherwise, such an animal is merely a carnivore or part of the obscure mortality estimates. Elaborate models of predator-prey relations have been created. In many cases, the abundance of prey, the food supply, and the energy costs of getting it determine or influence the predator population. There is ample room for work on managing raptors and other carnivores. This will involve prey management, working with a "herd of mice," naming deer as lion food, and working with some populations of rabbits as fox forage. To have certain animals, you must have their specific foods and more. You must have secondary foods. Almost all carnivores are opportunists; they must be for they live at the entropic edge. Most models do not work well for predators for they do not include the large and varied prey base of most forest stands. A rabbit-fox model ignores the reality of the fox that eats fruit, nuts, snakes, mice, frogs, birds, and eggs. Few modelers have done large, multispecies models at all well. They are very large, very tedious to create, and very difficult to run because after the 20th guess at an input, the reality of the situation hits - its unreality. See CAP9091.

So much of the literature of animal ecology relates to predation that it is difficult, risky, to ignore or minimize it. All animals forage; some forage on moving things. As a group these things are prey, a special kind of forage, relatively low in energy (when equated to the total system inputs) and relatively expensive of energy to catch them. It is difficult to estimate one population; to understand predators and thus prey, the faunal system manager must work with two things known with low probability - thus the product rule applies. Less than 70% confidence in both systems suggests the manager can do as well in decision-making about them with a fair coin.

A "predator" is a pejorative word. Otherwise, they are simply carnivores or, more likely in the forest, omnivores. Behavioral patterns are easily set in individual animals, less so in populations. An animal with a "bad" habit (like eating chickens) can usually be removed or excluded without classifying all animals of a similar species or life group as predators and subject to control efforts.

Evidently predation is a mortality factor. Predators' behavior shifts with the available food supplies and the cost to get it. In one season a species may eat only one animal group because it is abundant and easily taken. In the next year the same animals will develop an entirely new feeding strategy and will learn and pass on that strategy to learning offspring. Only long term, very intensive studies will lead to clear understanding of this phenomenon. It will be found, at high cost, that animal groups are unique; group behavior is in response to prey and past learned behavior; prey respond to faunal spaces, the"faunal surround"; these are only predictable within their broad bands; each area is unique and... Thus, predator-prey relations cannot ever be well predicted at a scale of managerial relevance.

Predators affect populations; predators may "regulate" some populations in some years ... but not every year. Other factors operate on prey and on predators. Sufficient control of"experiments" can never be gained to work this out. It is not surprising that"theories" change from year to year in study areas. New words are needed to explain what occurred last year in the same area.

One of the best and most simple ways to gain insight into predator-prey relations (expecting only gross, broad-band insights) is to use a phase plane (see Fig. 8.8). The predators are plotted as a function of prey, either last year or in the previous 2-3 years (say 3 years ago or the sum of the past 3 years). The pattern expected is one of a spiraling line, converging to some approximate central region. The circle resulting is a look down the center of a coil like an irregular-edged metal spring; the third dimension is time. Removal of predators allows limited prey to expand; prey numbers as food base expand; predators increase production and the population rebounds; more prey are eaten, thus a decrease. A natural"balance" is really a tight spiral, a small helix, isomorphic in the larger context with genetic structures. The coil over the years for a population in an area is variable and unlikely to be permanent due to changes in the ecological stage (age) of the forest.

Keith (1983) provided a simple approach to computing the number of prey needed per predator to allow the population to remain stable. He discussed wolves and deer but the concept is general and can provide insights into what is going on in a predator system. The relations are:

N = K /((r-1) (1+H))

where
N = prey per predator in the spring before births
K = the average number of prey killed per predator annually
r = potential finite rate of increase of prey
H = proportion of the annual increment of prey (e.g., deer) removed by hunting or other mortality factors.

Hunting and predation are assumed additive, not replacements for other mortality. The tonic of "compensatory mortality" needs much work. The math models, to date, are more elegant than the field work.

Where K may be about 15, r may be as high as 1.4 for unhunted and low-predator populations, and H may range however the manager may desire it (e.g., 0 to 0.4). These variables are easily manipulated in CAP9091, the results show desirable prey-to-predator ratios. The effect of lowering r (due to land use, forage, toxicants, etc.) has a profound effect in the desired prey-to-predator ratio. A change of 0.1 to 0.3 can cause a massive change (many times) in the ratio, even when hunting or other mortality (e.g., control efforts) are constant. Simultaneous change (reduction) in r, H, and modest fluctuation in K can have devastating effects on prey. There are low chances for an equilibrium occurring (a tight spiral in Fig. 8.20) when r is low.

Half of the applicants to wildlife graduate school programs include an interest in predator-prey relations in the list of topics in which they are most interested. Interesting indeed, but for the forest faunal system manager, it is a special topic of low importance. It is dominated, out competed, by the more general concept of survival forces.

Cycles

Fig. 8.20. A phase-plane diagram can provide insight into the relation of predator and prey. Using time (see inset) as the third dimension can suggest the helixical dynamics of populations around an equilibrium. The manager can usually change the central tendency of this movement to a point close to his or her objectives.

Evidently populations fluctuate. Whether they fluctuate regularly remains an interesting area of study. For the manager in an integrated vertebrate pest damage management system or a game-harvest-maximizing system or for someone contemplating replanting a forest, the question is simply: what will the population be next year? If the species is cyclic, then reduction may be natural and control costs small indeed. Hunting seasons need not be reduced in order to control losses in birds when they will decline anyway and hunters predictably will reduce their efforts because of reduced returns. Cycles are predictable recurrences of peaks and troughs in population density. Given the difficulty of estimates already described, open questions remain about which populations, if any, are cyclic. There is debate over whether cycles should have equal amplitude but this seems too constraining. At least the periods appear about equal. The question remains then: how much difference from normal (?) or average constitutes a peak? The periods are not invariable; the amplitudes are variable. Yet there has been abundant interest in the available data suggesting cycles. Errington (1954) [note the date] observed that the literature on cycles was voluminous and that he doubted " ... that any one person could now truly master it in a lifetime, even if ... competent in all of the fields of science contributing." Since then little more than the volume has changed. Doubt remains about existence of large wild animal population cycles. Where they appear to exist, there are people who seek the causative factor partially because they perceive region wide, simultaneous peaks or troughs in several species. Others perceive regular fluctuations, not simultaneous or universal, and search for precision in stating a multi-causal theory.

I do not know whether cycles occur in the species typically managed. I wish I did. I perceive that managers deal with systems more frequently in 3 to 10 year groups than the 30- to 300-year sets of records that are needed to "prove" cycles. Managers rarely can grasp the density estimation well enough to be confident. Highs and lows are fully expected. That their regular occurrence is 3 years, plus or minus 1 year, is meaningless. The 9-10 year cycle is more interesting (Keith 1963) and holds some managerial implication if it exists. Whether caused by sun-spot phenomena, predation, disease, vegetation or food, crowding stress, or random phenomena now seems almost irrelevant. The variability in the population components far exceeds the variability perceived; the ability to control differences in perceived benefits is very great. The species identified which might have cyclic population expression are few. My perception is that the evidence for these phenomena is from large regions. It is a cumulative phenomenon of the forage energy available to specific herbivorous groups, followed closely by prey. The phenomenon must be explained by aggregative phenomena of succession over thousands of tracts or stands. It is probably a function of the curvilinear expressions of forage available over time typically described as "succession." These curves often have a pi component, not unlike the 3.14 or 3-pi relations seen in forage succession curves. These have been mentioned in Chapter 7.

In understanding managed populations we come to see the details to which analyses may go, but also the oppressive combined variability, the limitations to gaining density estimates, and the inability to perceive what population levels we really want. This overall perception makes cyclic phenomena almost irrelevant, suggests limited meaning, if any, to "competition" between wild populations, at least any that cannot be easily regulated. It reduces problems of predator management to food production, thereby making prey of no more importance to a forest manager than a grain-food patch for game birds or a small, second-decimal contribution to a mortality rate estimate. Worse yet, contributing to uncertainty is a lack of discussion of feedback mechanisms, of how excessive densities of animals reduce food supplies, thus natality rates, thus future populations, and the role of predation in suppressing densities that are followed by natality. Excessive prey can increase carnivore natality that produce "boom" populations that can utilize the available resources. The profound needs are for the forest faunal system manager to master the profound factors discussed in this chapter; to master the variability within these factors, especially to exercise statistical control over them through using abiotic factors as managed in geographical information systems, and then to take more predictable managerial control. The forest system managers, and those who begin to comprehend geologic time (CAP625), know better than anyone that systems are created by cutting and planting or reseeding to last for 30 to 200 years. Once internalized, the annual perturbation means little. Even a cycle, if it exists, is limited and area specific, and there is little that can be done to fill the troughs or cut the peaks. The forest faunal system manager needs to understand populations to learn what actions to take as well as those not to take and those to prevent being taken in order to thwart population entropy and achieve human objectives.

This Web site is maintained by R. H. Giles, Jr.
Last revision: May 25, 2001.