One of the greatest promises of individual-based ecology is the ability to incorporate more realistic mechanisms in our models, thereby making them more general and more capable. Mortality is, of course, an extremely important mechanism in ecology. Mortality is important not only because it is a fundamental driver of population dynamics but also because, as ecologists now widely acknowledge (e.g., Peacor and Werner 2001; Preisser et al. 2005; Verdolin 2006), it is a strong driver of behaviors that trade off risk and other elements of individual fitness such as growth and reproduction. The most general and noncontroversial “first-principles” assumptions on which we can base models include that individual traits, including inherent behaviors, have evolved because they convey fitness, and that survival to reproduction is critical to fitness.

Representing how mortality risks vary (over space or time, or among individuals) is important for at least two kinds of model. First are models of the effects of changes that strongly affect risk; examples include land use changes and reintroduction of predators (both illustrated by Ganz et al. 2024). Second are models that include risk-avoidance behavior. If we want models to implement the assumption that behavior acts to increase fitness by avoiding mortality, the models must represent variation in mortality risk: model individuals cannot reduce risk if risk is the same in all places and at all times. Further, risk needs to vary at spatial and temporal scales relevant to behavior. If we model how animals select among habitat patches and how habitat selection varies seasonally, with patch selection dependent in part on predation risk, then we need to represent how risk varies among patch types and over seasons. If we want to model how animal behavior changes diurnally, then we need to represent how risk varies over the daily light cycle.

To address such purposes, individual-based ecology has the challenge of developing methods for quantifying how risk varies and for representing that variation in our models.

How individual-based models (IBMs) can represent behavior as fitness-seeking tradeoffs among risk, growth, and other elements of fitness has been addressed to some extent (e.g., chapter 5 of Grimm and Railsback 2005; Railsback and Harvey 2020), but there is little existing literature on general, mechanistic models of how risk varies with characteristics of habitat, time, and the individuals at risk. To get a general idea of how IBMs typically represent risk, we reviewed the IBMs in three recent volumes (499–501) of the journal Ecological Modelling, including the older models (published before 2009) described in the supplemental materials of Grimm et al. (2025). Of the 15 IBMs that included mortality, almost all represented multiple causes of mortality, e.g., unspecified “background” mortality and starvation when growth was low. However, only one (Anders et al. 2025) represented one cause of mortality that depended on multiple drivers: tree death from stress related to multiple climate variables, modeled via logistic regression on field data. We conclude that many IBMs represent mortality as multiple risks that each vary with a single variable, but few models explicitly represent how a risk such as predation depends on multiple variables.

While little of it addresses general, mechanistic modeling, there is extensive empirical literature on spatial and temporal variation in predation mortality and risk, especially in wildlife and livestock subject to predation by large carnivores (reviewed, e.g., by Prugh et al. 2019). Much of this literature uses statistical methods such as logistic regression on habitat variables observed where predation did and did not occur, to model how the probability of mortality varies (reviewed by Miller 2015). Other observational approaches are to (a) use tracking data on prey individuals and estimates of when they were killed to estimate predation risk in different habitat types (Ganz et al. 2024), (b) track the predators and observe predation events (Gervasi et al. 2013), and (c) develop resource selection functions for both predators and prey to estimate how the probability of predators encountering prey varies with habitat (Hebblewhite et al. 2005; Hebblewhite and Merrill 2007).

Such observational study methods and literature have limited value for the individual-based modeler. Studies that examine only mortality events cannot distinguish risk (as we define it below) from the confounding effect of prey exposure (the number of prey killed by predators is a function of both risk and number of prey). The literature and methods are generally limited to large terrestrial predators and prey, which are particularly easy to observe. Because these studies are purely empirical, their results are of questionable value for modeling other systems or novel future conditions. Observational studies also tend to have limited temporal resolution because it is generally difficult to determine precisely (e.g., between day, night, or twilight) when a predation event occurred. And these studies are of course expensive and time-consuming. However, observational studies provide valuable clues about general mechanisms (e.g., differences between wolves and cougars in predation success in dense vs. open habitat, and in willingness to hunt near humans; Atwood et al. 2009; Ganz et al. 2024).

Our objectives are to review concepts and mathematical methods that can be used to model how mortality risks vary over multiple dimensions in IBMs and other mechanistic ecological models. We specifically address models that treat one or more risks as explicit functions of multiple characteristics of habitat, individuals, or time. However, we do not address models in which mortality emerges from direct interactions among individuals, e.g., by representing adaptive predators or contagious disease spread.

We also discuss ways in which such methods can be supported by multiple kinds of information, from natural history knowledge, mechanistic understanding, and empirical experiments of several kinds. Our methods are especially applicable to systems and species for which reliable empirical information on risks is difficult to obtain, and for models that benefit from being more mechanistic and general because those characteristics make them useful for predicting responses to novel conditions under which strictly empirical representation of risk would be unreliable.

We focus, but not exclusively, on predation risk to mobile animals that use risk avoidance behaviors such as selecting when and where to feed. Our experience is primarily with modeling salmonid fish, specifically the “InSTREAM” family of IBMs for predicting effects of river management on trout and salmon populations (e.g., Railsback and Harvey 2002; Railsback et al. 2023). These models represent multiple kinds of mortality that are affected by managed variables such as flow and temperature. Simulated risks include two kinds of predation, starvation and disease, and extreme temperatures. Reliably modeling how such risks vary with habitat and among individuals is essential for understanding how management affects individual behavior and fitness and, therefore, population dynamics. InSTREAM assumes individual fish select when and where to feed (or hide) as an adaptive tradeoff between risk and growth (Railsback et al. 2020). However, the methods we present are also applicable to other kinds of risk such as disease, human harvest, and extreme weather or habitat conditions. The methods also seem useful for modeling such risks to plants as herbivory, lack of resources (water, light, nutrients), pest infestation, and disease.

Traditional equation-based ecological models (e.g., the Lotka-Volterra equations) represent mortality via a rate parameter d, the fraction of a population that dies within a specified unit of time (Haefner 2012). However, in an IBM mortality is not a population-level rate but an individual-level event: every time step, each individual either survives or dies. A very simple way to model mortality as an event is using the variable “risk” R as a probability of mortality within a specific time period. We can set the value of R by assuming it equal to an observed mortality rate, but it is important to remember that when we model mortality of individuals in this way, R is a probability, not a rate.

(Here, we generally assume 1 day as the time period. We also use “risk” as a general term for mechanisms that could cause mortality, e.g., the risk of heart attack. The context should make it clear when “risk” is used in this general sense or specifically as a probability of death, R.)

The term “mortality” also has general meanings—the phenomenon or rate of death among individuals—and a specific meaning, the death of an individual. Mortality of an individual is typically represented in IBMs as an event that occurs randomly with probability R of occurring within one time period.

Studies and models that explicitly represent both predators and prey often follow the convention (e.g., Holling 1959) of decomposing R into separate probabilities of (a) a prey individual encountering a predator and (b) the predator successfully killing the prey when an encounter occurs. Similar decomposition can be applicable to other kinds of mortality; disease, for example, is often represented via separate probabilities of exposure to, infection by, and succumbing to a pathogen. While such decomposition is helpful for understanding and quantifying some risks, here we do not include it when discussing predation because it is unnecessary when predators are not represented as explicit individuals.

Models can be simpler to describe and implement when mortality is represented via survival S, the probability of surviving a specific time period, so S = 1.0 – R. Under the very common (yet often unstated) assumption that mortality risk in any time period is independent of risk in adjacent periods, the probability of surviving any t time periods is simply S^t, which is especially convenient for models with time steps of variable lengths. Our trout model simulates four time steps representing the light phases of each day (Railsback et al. 2021); the length t (fraction of 1 day) of each phase varies with date. On each time step we model survival of each kind of mortality as a daily probability S_X, and then determine mortality each time step using S_X^t as the probability of surviving.

Modeling risk as a survival probability is also convenient because the probability of surviving multiple, independent, kinds of mortality (discussed below) is calculated simply as the product of the survival probabilities for each mortality type.

Even though we typically evaluate risk and survival as daily probabilities, understanding and making good modeling decisions requires thinking about risk over longer time periods. Seemingly minor differences in daily survival result in large differences over meaningful future periods.

The widespread misuse of the so-called “µ/f rule” (or “µ/g”) provides a good example of the importance of time for understanding risk. Gilliam and Fraser (1987) derived, for a highly simplified system, that long-term fitness for an individual is maximized by selecting the behavior that minimizes the ratio of risk to food intake. Presumably because of its simplicity, this “rule” has been used to represent tradeoff decisions in a variety of models. When we think of daily probabilities and rates, the “rule” may seem reasonable—a small increase in risk, say 10%, is the price for a corresponding increase in food intake. However, when we think about future survival this tradeoff seems less reasonable. If risk is low, e.g., daily S is 0.998 so the probability of surviving for 30 days is 94%, a 10% increase in daily risk reduces the probability of surviving for a year by only 7%. But if risk is already high, e.g., S = 0.98, a 10% increase in risk reduces the probability of surviving for a year by 50%. The tradeoff of 10% gain in food for 10% increase in risk is a bad one when risk is already high and behavior should instead emphasize reducing risk. Looking at the long-term consequences of changes in risk makes it clear that the “µ/f rule” cannot provide good tradeoff decisions across wide ranges of risk.

When estimating survival parameters for models, it helps to think about survival over a meaningful time period and then calculate the corresponding daily probabilities. For example, we might estimate that an animal in a particularly risky habitat has a 20% chance of surviving for a month. The corresponding daily survival is: $S=0.2^{\left(\frac{1}{30}\right)}=0.948$. Another approach is to estimate a median survival time (t_m) and calculate daily survival probability from it: $S=0.5^{\left(\frac{1}{t_m}\right)}$.

In some models it is important to distinguish among risks that individuals do vs. do not perceive and respond to. Human harvest, vehicle collisions, disease, natural toxins, pollutants, and introduced predators are example risks that a modeler might assume animals are naïve about and do not avoid via behavior. Such risks can be represented by modeling them exactly as the other risks while assuming that the individuals do not consider them in risk-driven behaviors. For example, Ayllón et al. (2018) added angler harvest to a trout IBM in which individual trout select habitat to trade off growth and predation risk. The risk of mortality via angling was assumed to vary with trout size, season, and angling regulations, but that variation was not considered by model trout in their behavior.

Here we list some of the “dimensions” in which mortality risks can vary, ending with a summary of those dimensions in our salmonid models. These dimensions are variables or groups of variables that risks often vary over.

The first step is determining which types of risk to model separately. We assume that the modeler has carefully determined what risks need to be in a model to meet its intended purpose. “Pattern-oriented modeling” is a powerful strategy for doing so, thoroughly illustrated by Grimm and Railsback (2005, 2012). As Table 1 illustrates, we need to model risks separately if they are driven by different variables (e.g., thermal stress is driven by temperature while starvation and disease are driven by an individual’s weight relative to its length), or if they are driven by the same variables but in different directions (e.g., increasing depth makes a small trout less vulnerable to terrestrial predators but more vulnerable to fish predators). Another reason to separate risks into different types is to allow model results to indicate how much mortality was caused by each risk: if we want to observe how many simulated elk are killed by wolves vs. cougars, then we need to model these two predators separately.

There are of course costs of using more types of risk: more assumptions and parameters, and more computations. While traditional modelers associate more parameters with higher uncertainty, a more-resolved representation of risk could reduce uncertainty by supporting more realistic behavior and results and by making more empirical evidence useful for representing risk.

Next, for each type of risk we need to model how survival probability S varies with one or more selected variables. Selecting the variables that drive each risk is similar and related to the process of selecting types of risk: variables should be included if they are considered essential to the model’s purpose or—via pattern-oriented modeling—essential for establishing the model’s credibility. But a variable might be considered essential for a model only because it is a critical driver of risk.

We discuss four ways to model how S depends on selected variables, but focus on the third and fourth, which we find most useful and novel.

Above (Conventional approaches to variation in risk) we identify examples of studies that modeled risk statistically, from several kinds of data, and discuss limitations of such approaches. Logistic regression is an appealing analysis framework because it estimates how the probability of an event—mortality—depends on multiple variables. However, modelers must be aware of differences between observed mortality rates and the survival probability variables in their models, especially for models containing risk-avoidance behavior. A model parameter representing S in the absence of risk avoidance behavior might be poorly estimated by data on actual mortality events, because actual mortality rates depend on prey abundance and behavior as well as the underlying survival probability.

Continuous univariate models of S

Risks assumed to depend on only one variable can be modeled as continuous functions that are either estimated from observations (as in Option 1) or designed by considering a variety of information. Such univariate functions could have any shape as long as they produce values between 0.0 and 1.0. We find two forms especially useful.

Logistic curves are useful for representing nonlinear effects of a variable on survival (Fig. 1), for several reasons. First, they produce S values (the Y axis) ranging from 0.0 to 1.0 as the driving variable (X axis) varies over its entire range. They are also good at representing variables that produce high and low survival over wide ranges but sharp changes in survival between those ranges, which are quite common. Finally, when the right kind of observations are available, logistic curves are readily fit to data via logistic regression.

Figure 1. 

Example logistic curve for S, representing trout survival of high-temperature stress. The round symbols represent observations from two laboratory studies of daily survival at temperatures of 24, 26, and 28 °C. The solid curve is a logistic function fit to the three observations, the model of daily S. The square symbols indicate the two parameters that define the logistic curve: X_0.1 and X_0.9 are 30.2 and 25.8 °C. The dotted curve shows 10-day survival probability, equal to S¹⁰.

The equation for a logistic function is:

$Y=\frac{\exp \left(Z\right)}{1.0+\exp \left(Z\right)}$ Eq. 1

where Z is a function of the X value:

$Z=A+\left(B\times X\right)$. Eq. 2

We find it convenient to define A and B, and therefore the shape of the logistic curve, via two model parameters, X_0.1 and X_0.9, which are the X values that produce Y values of 0.1 and 0.9 (Fig. 1): B = -4.3944 / (X_0.1 – X_0.9) and A = -2.1972 – (B × X_0.1).

Because of the exp functions in equations 1 and 2, logistic functions are particularly vulnerable to variable overflow/underflow errors, which occur when the computer tries to produce a floating-point number larger or smaller than it can handle. The ranges of values causing errors depend on the software platform and sometimes the hardware. These errors can be avoided while providing the ability to evaluate logistic functions over wide ranges by adding code that sets Y to 1.0 when Z is greater than (e.g.) 35 (corresponding to Y = 0.999999999999999 in 64-bit Excel) and to 0.0 when Z is less than -200 (corresponding to Y = 1.4E-87).

Linear models of survival can be useful for chronic risks that persist over long times and cannot be alleviated rapidly. We use a linear model of how survival of starvation and disease depends on an individual’s energy reserves, in models in which energy reserves can change relatively little in one time step. If we define an individual variable energy-deficit as the fraction by which energy reserves are below those of a healthy individual (so, e.g., energy-deficit = 0.2 means the individual’s energy is 20% below a healthy value), we can model the daily probability of not starving S_S as linear: S_S = 1.0 – (P_S×energy-deficit) where P_S is a parameter. Higher values of P_S cause survival to decrease more rapidly as energy decreases.

This kind of linear model produces a gradually decreasing probability of surviving prolonged periods of risk, e.g., when energy-deficit is constantly positive or increasing steadily due to constant weight loss (Fig. 2).

Figure 2. 

Example linear survival model, representing survival of starvation and disease S_S as a function of energy deficit. The solid lines indicate the value of S_S when P_S is 0.2, when energy-deficit (left) is constant at 0.2, and (right) starts at 0.0 and increases by 0.01 on each of 30 days. The dotted lines indicate the cumulative probability of surviving from day 0.

Univariate models can be fit to data, including data from multiple sources, or even to a set of assumed values. Both linear and logistic functions can be fit via regression when suitable data are available, and via other techniques when data are not suitable for regression. For example, the logistic parameters X_0.1 and X_0.9 illustrated in Fig. 1 were fit to the three observed survival rates by using Excel’s Solver to minimize the sum of squared differences between observed and logistic-curve values.

Multivariate models using “survival increase functions”

None of the previous three methods for modeling S have all of the following characteristics that we find essential for modeling some kinds of risk, especially predation:

• S can be modeled as a function of multiple variables (e.g., how a fish’s survival of predation by birds varies with fish size, depth, and light intensity);
• Variables can be added to the model, or the relation between one variable and S can be modified, without having to re-fit or re-calibrate the entire model of S;
• The relations between variables and S can take different forms, including continuous and discrete relations (e.g., the effects of fish size and depth on predation risk are continuous functions but the effect of light intensity is represented as discrete values for day, night, and twilight);
• The value of S (i.e., the overall intensity of predation) can be calibrated easily by changing one parameter; and
• The relations between each variable and S are easy to see, understand, and fit to data or assumptions.

Simply fitting a multivariate equation for S would not provide these characteristics, so we developed the following method for modeling complex survival probabilities. It uses separate functions to represent the effect of each variable, with those effects then combined into a survival probability. As we discuss below, this approach assumes that variables driving S have independent effects.

First, we identify a minimum survival probability S_min, which is a model parameter. The value of S_min represents survival under the least-safe conditions.

Second, we estimate “survival increase functions” (SIFs) for each variable (of model individuals, their habitat, or anything else) that affects survival. The SIFs produce “survival increase” values that range from 0.0 to 1.0 and represent the degree to which the variable increases survival probability: a SIF value of 0.0 provides no increase in survival and a value of 1.0 makes the individual completely safe (S = 1.0). However, SIFs have no other limitations on their form or origin: each can be a different function type developed from different information.

The function types described above as univariate models of S are also useful as SIFs. Most of the SIFs we use are logistic curves, but we also use discrete functions to represent the effects of boolean (true-false) or categorical variables. For example, a SIF for use of hiding behavior can simply be: survival increase is 0.8 if the individual is hiding and 0.0 if not. A SIF for the effect of light on risk could simply be: survival increase is 0.0 in daytime, 0.6 during twilight, and 0.9 at night.

A SIF can also be modified so that its value never reaches 1.0—no values of its variable make individuals completely safe. For example, a model of predation survival could include a variable representing vegetation density, and assume that survival probability is higher when vegetation is denser but some predators can be successful in even the densest vegetation. We can model that effect with a SIF that uses a logistic curve limited to values less than 1.0; such a curve only requires one additional parameter for the maximum survival increase, which all logistic curve values are multiplied by.

Example SIFs (for the risk of terrestrial predators on trout) are illustrated in Fig. 3. The function for depth (panel A) is a logistic curve with X_0.1 and X_0.9 equal to 20 and 150 cm and a maximum value of 0.8: even trout in the deepest water are still vulnerable to diving predators such as otters and mergansers. Panel B is the length function, a logistic curve with X_0.1 and X_0.9 equal to 6 and 3 cm: very small trout are less visible and less valuable to predators, and the largest trout are still vulnerable to many predators. The light function is also a logistic curve, with X_0.1 and X_0.9 equal to 50 and -10 W/m²: only light levels at or below those characteristic of twilight reduce risk. The SIF for use of hiding cover (Panel D) is a discrete function: the survival increase value is 0.8 if a trout is hiding in concealment cover, and otherwise 0.0.

Figure 3. 

Example survival increase functions, for variables affecting terrestrial predation risk to trout in Table 1. The round symbols indicate survival increase values at (A) depth = 100 cm (survival increase = 0.5), (B) length = 6 cm (0.1), and (C) light intensity = 20 W/m² (0.5). Survival increase for fish hiding in concealment cover (D) is 0.8.

The third step is to combine the SIFs into a value of S, using a method (Railsback et al. 2023) in which each SIF contributes to reducing risk, so survival depends on all such variables. We model the interaction among SIF values (F_i where i indicates the functions, e.g., panels A–D in Fig. 3) by treating each F_i as a probability and calculating the joint probability of surviving all of them:

$S=S_{min}+\left(1-S_{min}\right)\left(1-{\Pi }_{i=1}^{i=n}\left(1-F_i\right)\right)$. Eq. 3

Using the example survival increase values of Fig. 3, the product term in Eq. 3 is: (1–0.5)(1–0.1)(1–0.5)(1–0.8) = 0.045. If S_min is 0.9, then S = 0.9955. Using this method, all the SIFs affect S but S is most sensitive to those with highest values (Fig. 4).

Figure 4. 

Example values of S from Eq. 3 when it includes the depth and light SIFs of Fig. 3. S_min is equal to 0.9. When light intensity is low, S is most sensitive to light; otherwise, S is most sensitive to depth.

One limitation of this method is that it does not represent interactions among the ecological factors affecting risk, e.g., the relation between depth and S depending on the value of light intensity. Such interactions could of course be added to a model, if the benefits of additional realism appear to outweigh the costs of substantial additional model complexity.

The final step in using this method is to estimate a value for S_min. We typically do this via calibration of the full model, searching for S_min values that produce reasonable model results. In many models, values of such survival parameters are found by fitting results to observed or assumed values of abundance or survival rate. However, models with tradeoff behaviors that relate (e.g.) growth to risk may also produce growth and size results that also respond strongly to S_min. For such models, it may be necessary to calibrate S_min simultaneously with parameters that drive growth.

If a model includes more than one type of risk that uses this method for S (e.g., predation by terrestrial animals and fish, in Table 1), it may be possible to estimate each value of S_min relatively independently, if the two types of risks mainly affect different life stages or sizes of individuals. In our trout model, only juvenile trout are vulnerable to predation by other fish, so we can estimate S_min for that risk using observed juvenile survival rates, and then estimate S_min for terrestrial predators using overall population abundance.

The authors have declared that no competing interests exist.

No ethical statement was reported.

No funding was reported.

Conceptualization: SFR, BCH. Writing – original draft: SFR, BCH.

Steven F. Railsback https://orcid.org/0000-0002-5923-9847

Bret C. Harvey https://orcid.org/0000-0003-0439-9656

All of the data that support the findings of this study are available in the main text.