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Article review by Geoffrey Peterson

Article: Mills, N.J., Getz, W.M. "Modelling the biological control of insect pests: a review of host-parasitoid models." Ecological Modelling, 92:121-143, 1996.

Unless specified, all information comes from this source. Full references to specific peoples work appear at the end of Mills and Getz article, which can be found here. Also, any mathematical formulas appear as LaTeX script for when/if LaTeX will be successfully incorporated.


In this article, the authors present a review of various models that have been developed over the years to model the interactions of parasitoid insect populations with their respective host populations. Specifically, it discusses these models in regards to their relation to classical biological control, which is the practice of introducing a natural enemy of an insect pest with the goal of reducing the insect pest's population to a level that is no longer dangerous to the local region. As it does not chemical or non-natural means of reducing exotic species, biological control is favored method of pest control and has been used to great effect in the United States and in the rest of the world. However, there are examples of biological control agents that become invasive and have devastating impacts on non-target species.

Basic Framework Models

The population dynamics of biological control has been developing for many decades, and host-parasitoid models have been favored due to the following three assumptions that can be made:

  1. Closed system: Because most insect pests have a specific parasitoid associated with it, we usually do not have to worry about external populations. Thus, we consider the system to be closed and can disregard the effect of other populations.
  2. Equivalent generation times: Again due to the specificity of the host-parasitoid interaction, we can consider the generational times between the groups to be equivalent as the parasitoid's breeding period will depend on the host's breeding period.
  3. Ignore/simplify age structure: Hosts are attacked by adult female parasitoids, so the age structure of the hosts and parasitoids should not affect the overall interactions. Thus, we can either ignore or simplify the age structure to make them easier to handle.

Of course, these assumptions may not apply to all types of host-parasitoid interactions, and when we ignore some of these assumptions, we can obtain interesting refinements to the model. There are two major categories for host-parasitoid models which depends on how generations are measured in time. These two categories are discrete generations or continuous generations.

Discrete Time

When the generations of the hosts and parasites do not overlap, the generations are measured in discrete times, usually by year or breeding season. These kinds of systems where generations are distinct are common in temperate regions of the world, but occasionally, they can be found in tropical regions where parasitism create generational cycles within the overlapping generational cycles of the host. The framework for this model is a coupled set of difference equations:

  • N_{t+1} = d(N_{t}) N_{t} f(N_{t}, P_{t})
  • P_{t+1} = c N_{t} (1 - f(N_{t}, P_{t}))

where N_{i} and P_{i} are the host and parasitoid populations at generation i, (N_{i}) represents the per capita net rate of increase for the host population (i.e. proportional growth rate), and f(N_{i}, P_{i}) is the proportion of the host population that are NOT attacked by the parasitoid population. c is the average number of parasitoids that emerge for every parasitized host, and it includes numerical characteristics such as the average number of eggs laid in each host, survival rates of parasitoids inside attacked hosts, and gender ratios of parasitoid adults that emerge.

Continuous Time

When generations overlap, we model the populations in continuous time. Originally developed by Lotka and Volterra to model predator-prey relationships in vertebrates (Some general information on Lotka and Volterra model can be found here), the framework is a coupled set of differential equations:

  • dN/dt = g(N) N - h(N, P) P
  • dP/dt = \gamma h(N, P) P - \delta P

where N is the host population (which encompasses all stages of development though usually only the adults) and P is the number of adult females in the parasitoid population. g(N) is the per capita rate of increase of the host population, \gamma is the called the "conversion efficiency" of hosts to parasites, which relates to the number of parasitoids that emerge from a parasitized host, and \delta is the per capita death rate of parasitoids. It should be noted that there is no specific term for the number of hosts that die independently of parasitoid attacks, but due to the arbitrary definition of g(N), such a term can be incorporated based off the specificity of the host-parasitoid interactions. Finally, h(N, P) refers to the "functional response" of the parasitoid, which is discussed in the next subsection.

Functional Response

  • First used by an ecologist named Holling in 1959 to describe the relationship between shrews and deer mice that feed on sawfly cocoons, the functional response expression is usually understood as "the rate at which an individual consumer extracts resources as a function of resource density" (Mills, pg. 123). Essentially, the functional response in regards to host-parasitoid models corresponds to the proportion of hosts that are attacked by parasitoids during a given time period.
  • For continuous models, this function appears simply as h(N, P). For discrete models, Mills and Getz argue that the functional response corresponds to the quantity h(N_{t}, P_{t}) = N_{t} (1 - f(N_{t}, P_{t})) / P_{t}. This quantity in the discrete case comes from calculating the mean number of encounters per host, which is used to set f(N_{t}, P_{t}) to the probability of zero encounters occurring between a host and a parasitoid using either a Poisson or negative binomial probability distribution.
  • For both continuous and discrete models, the functional response equation can appear as one of three basic types, depending on the relationship of the functional response equation to the host population abundance, N. To maintain generality, the functional response equations depend on two arbitrary positive constants u and v.

Type I: Linear in N

  • Continuous: h(N, P) = uN, for N < v/u, and h(N, P) = v, otherwise
  • Discrete: h(N_{t}, P_{t}) = N_{t} (1 - exp(-u P_{t})) / P_{t}

Type II: Asymptotic in N

  • Continuous: h(N, P) = uN / (v + N)"
  • Discrete: h(N_{t}, P_{t}) = N_{t} (1 - exp(-u P_{t} / (v + N_{t}))) / P_{t}

Type III: Sigmoidal in N

  • Continuous: h(N, P) = uN^2 / (v^2 + N^2)"
  • Discrete: h(N_{t}, P_{t}) = N_{t} (1 - exp(-u N_{t} P_{t} / (v^2 + (N_{t})^2))) / P_{t}

(A more qualitative description of Type I - III can be found in the Background Section of APPM4390:Three_Species_Food_Chain_Modeling, by Dowling, Klopfenstein, and Koepke)

Early Models

Mills and Getz presents a number of early models for host-parasitoid models, classifying them by time-steps (continuous or discrete) and by functional response types. Although modern models are naturally more complex and more accurate, these models highlight a number of interesting features, which are summarized here.

Thompson Model

  • N_{t+1} = \lambda N_{t} exp(-\beta P_{t} / N_{t})
  • P_{t+1} = \lambda N_{t} (1 - exp(-\beta P_{t} / N_{t}))
  • Developed by W.R. Thompson in 1929, this host-parasitoid model is a discrete time model with type II functional response. \lambda is the per capita net rate of increase in the host population, and \beta is the mean number of eggs laid per parasitoid. It was the first to use a probability function (the exponential comes from a Poisson distribution) to describe parasitoid attacks, and it introduced the assumption that attack rate should be limited by egg production.
  • Ignores issues with survival rates and parasitoid gender ratios, and indicates unstable interactions between host and parasitoid. Because no non-trivial equilibrium exists, this model means that both populations will either decrease to extinction or increase without bound, and neither scenario is reasonable.

Nicholson-Bailey Model

  • N_{t+1} = \lambda N_{t} exp(-a P_{t})
  • P_{t+1} = N_{t} (1 - exp(-a P_{t}))
  • The most familiar and influential discrete-time model, it was developed by A.J. Nicholson and V.A. Bailey in 1935 and uses a type I functional response. The parasitoid attacks occur at random among host individuals, like the Thompson model, but the average attack rate is driven completely by the number of parasitoids instead of egg limitation.
  • Still shows unstable interactions, and both populations will experience divergent oscillations until the parasitoid population reaches extinction.

Holling Model

  • N_{t+1} = \lambda N_{t} exp(-a T P_{t} / (1 + a T_{h} N_{t}))
  • P_{t+1} = N_{t} (1- exp(-a T P_{t} / (1 + a T_{h} N_{t})))
  • A variation of the Nicholson-Bailey model, this discrete-time model with type II functional response was developed by C.S. Holling in 1959 under the hypothesis that the search for hosts are limited by TIME. T thus represents the lifetime of a parasitoid, and T_{h} represents the handling time for an individual host. It adds to the instability of the Nicholson-Bailey, but it is notable because it was the first to introduce time limitation in host-parasitoid models.
  • For more on Holling Models see Three Species Food Chain Modeling.

Lotka-Volterra Model

  • dN/dt = r N - a N P
  • dP/dt = \gamma a N P - \delta P
  • Although originally developed independently by A.J. Lotka in 1925 and V. Volterra in 1926 to model predator-prey interactions, this model can model host-parasitoid interactions on continuous time frames, using the linear portion of the type I functional response. It has become the basic framework for every host-parasitoid model that is measured in continuous time. Unlike Nicholson-Bailey models, Lotka-Volterra actually predicts stable oscillation cycles in parasitoid and host populations.
  • However, because Lotka-Volterra runs in continuous time, parasitized hosts remain vulnerable to parasitoid attacks, leading to a phenomenon known as "superparasitism" unless parasitoids can differentiate between parasitized and non-parasitized hosts, and the model can be made to incorporate the ability to differentiate between parasitized and non-parasitized hosts.

Refinements of the Models

From the framework and early models of host-parasitoid models, the authors present a number of refinements to the models, mostly using the Nicholson-Bailey and Lotka-Volterra models, to demonstrate how certain additions can affect the populations' overall stability and trajectory. These additions often reflect a specific limitation, or heterogeneity, inherent in the host-parasitoid interactions, and not all additions lead to sensible results. The analysis is quite extensive, so only the major rationales and results are presented here.

Density-Dependent Self-Limitation

To induce stability in the Nicholson-Bailey model, density-dependent self-limitation is added to the model. Originally, self-limitation was accomplished by raising either the host abundance, N_{t}, or parasitoid abundances, P_{t}, to the (1 - m) power, where m represents the severity of the density dependence. These power functions create damped oscillations in the trajectories, however, M.P. Hassell showed in 1978 that these functions retained unrealistic properties, which limited their use as general models. J.R. Beddington remedied this problem in the same year by using exponential functions of the logistic expression, switching the distribution of parasitoid attacks from random to aggregrated. The moderate success of this work has led to further studies to model the effect of host refuges from parasitoid attacks.

Spatial Limitations: Probability Distributions for Parasitoid Attack

Spatially speaking, host populations can avoid parasitoid populations and protect themselves from attacks using refuges. Refuges can take on a number of different forms, from simple hiding of host populations to the aggregrated distribution of parasitoid attacks, but the real goal of refuges is to create a specific proportion of the host population that is considered invulnerable to parasitoid attack.

The aggregrated distribution of attacks is the preferred method of modeling spatial heterogeneity, which can be added to Nicholso-Bailey by changing the probability distribution of the functional response from the zero term of the Poisson distribution to the zero term of the negative binomial distribution, and non-trivial stability within the system is obtained. This led to other models that included both density-dependent aggregration (DDA), density-independent aggregrative (DIA), and the so-called CV^2 > 1 rule that states the if the coefficient of variation squared exceeds one, then both forms of aggregration stabilize the interactions.

In continuous time models like Lotka-Volterra, density-dependence can be added by setting the per capita net rate of increase g(N) = aN + abA N^{x-1}, where a represents the attack rate, b represents the degree of aggregration, and A and x are positive constants related to the variance of host patch densities. However, no values for these constants create stability in the system, leading to the conclusion that neither DDA nor DIA have an effect on host equilibrium or stability.

However, this analysis leads to the "paradox of biological control" in both discrete and continuous time models. This paradox involves the existence of a trade-off between the stability of the system and the parasitoid's ability to successfully suppress the host density.

One important assumption that receives little attention is that parasitoids are search limited over all host densities, which may not be true. In fact, parasitoid attack rate is actually search limited only at low to moderate host densities, but at high densities, the rate is limited by the amount of egg production. This concept is actually confirmed by the authors using a discrete-time model that combines both search and egg limitation within the functional response equation.

Temporal Limitations: Age-Structure

One concern that is ignored by the Nicholson-Bailey and Lotka-Volterra models is the exact age stage at which hosts are parasitized. Normally, it is assumed that the models demonstrate the portion of the populations that are actually involved in the host-parasitoid interactions, and any other age stages are either host individuals that are invulnerable to parasitism or parasitoid individuals that are incapable of parasitism. In 1987, W.W. Murdoch fully examined the effect of an age structure by incorporating delay-differential equations in the Lotka-Volterra model:

  • dU(t)/dt = E(t) - M_{U}(t) - a P(t) U(t) - d_{U} U(t)
  • dA(t)/dt = M_{U}(t) - d_{A} A(t)
  • dJ(t)/dt = a P(t) U(t) - M_{J}(t) - d_{I} J(t)
  • dP(t)/dt = M_{J}(t) - d_{P} P(t)

Here U(t) and A(t) are the densities of immature and adult hosts with, J(t) and P(t) are the densities of juvenile and adult parasitoids, the d_{i} terms are the per capita death rates, E(t) is the daily density of eggs produced by adult hosts, a is still the attack rate, and M_{U}(t) and M_{J}(t) are the maturation rates of immature to adult hosts and of juvenile to adult parasitoids. The maturation rates are expressed using delay equations in terms of E(t), P(t), and U(t).

From the model, the authors conclude that an invulnerable adult host stage is stabilizing over the immature host stage, which makes sense as most host populations are more susceptible to parasitoid attack when they are juveniles. Thus, the stability of the entire model is dependent on the longetivity of adult hosts relative to immature parasitoids, but the trade-off between stability and host equilibrium density still exists.

Similar to spatial heterogeneity, temporal heterogeneity provides hosts with refuges from parasitoid attacks because of an asynchrony between host susceptibility stages and parasitoid foraging stages. For example, if the hosts are susceptible to attacks during a specific time of year but parasitoids only search out and attack hosts during some other period of time, the parasitoid will be less efficient in reducing the abundance of the host population due to this asynchrony. Time-delay differential equations similar to the ones used for age-structured models were used by K.J. Griffiths in 1969 and M. Munster-Swendsen and G. Nachman in 1978, leading to conclusions that temporal refuges due to asynchrony may have an important effect on host-parasitoid interactions.

Intraspecific Competition

One major assumption of host-parasitoid models was the existence of a closed system due to the specificity of host-parasitoid interactions. However, in real environments, it is conceivable that multiple species of parasitoids could prey upon the same population of hosts, creating intraspecific competition between the parasitoids, so biologists would naturally want to look at how a secondary parasitoid population changes the dynamics.

In discrete time models, the equations for the host, N and primary parasitoid, P, populations take the form of the Nicholson-Bailey model with density-dependent self-limitation:

  • N_{t+1} = N_{t} exp(r (1 - N_{t}/K)) (t + a_{P} P_{t}/k)^{-k} (1+a_{Q} Q_{t}/k')^{-k')
  • P_{t+1} = N_{t} (1 - (1 + a_{P} P_{t}/k)^{-k})

where a_{i} terms represent the attack rates from either the primary or secondary parasitoid, and k and k' indicate the amount of self-limitation in the parasitoid populations. As the primary parasitoid population, P is assumed to attack hosts independently of the secondary parasitoid, Q, forcing the secondary population to only attack hosts that survive attacks from the primary parasitoids. Thus, the equations for the secondary parasitoid depend on whether it is assumed that the two parasitoids occupy independent or identical niches in the ecological system:

  • Independent niches: Q_{t+1} = N_{t} (1 + a_{P} P_{t}/k)^{-k} (1 - (1 + a_{Q} Q_{t}/k')^{-k'})
  • Identical niches: Q_{t+1} = N_{t} ((1 + a_{P} P_{t}/k)^{-k} - (1 + a_{P} P_{t}/k + a_{Q} Q_{t}/k)^{-k})

The models indicate a secondary parasitoid population can persist in a given system with limited effect on the stability, and the host equilibrium density is reduced. If independent niches are assumed, the primary and secondary parasitoids together reduce the host density below what each is capable individually, but if identical niches are assumed, the parasitoid with the greatest attack rate always maximizes the reduction in host equilibrium density.

For continuous time models, the equations mirror those the delay-differential equations used in the age-structure analysis. However, under these equations, co-existence between the two parasitoids occurs only under limited conditions, and stability is again dependent on the duration of the invulnerable adult host stage. There are also considerations of which parasitoid is intrinsically superior to the other, can one kind of parasitoid discriminate a host that was parasitized by the other kind of parasitoid, and whether the attack rates favors the inferior or superior parasitoid. Even still, these considerations do not lead to great reductions of host density due to the intraspecific competition, so if hosts have an age-structure with an invulnerable adult phase, it is better to use a single parasitoid that is intrinsically superior to the other to maximize host density reduction.

Host Feeding

Another consideration for host-parasitoid interactions is the possibility that the parasitoid not only uses the host for parasitic egg production but also as food source. It is perfectly reasonable that parasitoids that use hosts to incubate and protect eggs would also feed upon the hosts, and it actually occurs quiet frequently among hymenopteran parasitoids. In the case of host feeding, a portion of the host population is being limited by the density of the parasitoid population, but instead of producing parasitoid eggs, this portion simply dies off.

This predatory interaction was first incorporated into a Lotka-Volterra model without age-structure by N. Yamamura and E. Yano in 1988, and host feeding appeared to stabilize the host-parasitoid model. However, it is unclear why this stabilization occurred. Lotka-Volterra models were initially designed to model predation, not parasitism, so the addition of host feeding could simply be accessing the stabilizing effects of predation in the Lotka-Volterra model. This would indicate that the parasitoid interaction is having little to no effect on the model. As a result, we do not know how host feeding truly affects continuous-time models at this time.

However, in 1989, N.A.C. Kidd and M.A. Jervis incorporated host feeding into a stabilized Nicholson-Bailey model without age structure using the following equations:

  • N_{t+1} = (R - m N_{t}) N_{t} exp(-a P_{t})
  • P_{t+1} = (1 - (1 / (log(N_{t}) + 1) ) ) N_{t} (1 - exp(-a P_{t}))

where R is the maximum growth rate of the population, m represents the degree of host density dependence, and a is still the parasitoid attack rate. This model indicates that host feeding had no effect on the stability of the system, though when age discrimination was added in which parasitoids feed on younger hosts and lay eggs inside older hosts, a limited degree of stability was observed.

The conflicting conclusions between the Yamamura-Yano model and Kidd-Jervis model was later addressed by C.J. Briggs, R.M. Nisbet, and W.W Murdoch in 1995. They used a Lotka-Volterra framework with a parasitoid population structure around egg load (the number of eggs stored within an adult female parasitoid) and found that host feeding did NOT effect stability. Thus, it is generally assumed that host feeding will have little to no effect on the host-parasitoid interactions.

Case Studies

In this article, these conclusions were evaluated in detail using case study data.

  • Winter moth. Photo courtesy of Colin Smith
    Winter moth: In the 1950s, winter moths were harming hardwood trees in eastern Canada, so a parasitoid was introduced from the moths' native Europe, which greatly reduced the abundance of winter moths. This reduction was most easily modeled at the time with the Nicholson-Bailey model and aggregrated parasitoid attack rates, but upon review of the data, the winter moth case study appeared to be more a matter of predation that parasitism because of the decline in unparasitized pupae in the soil.
  • For a more in depth look at the life cycle of gypsy moths see APPM4390:Kupsa
  • Cassava mealybug. Photo courtesy of the Consultive Group on International Agricultual Research
    Cassava mealybug & California red scale: More recent application of host-parasitoid interactions include the cassava mealybug, an insect pest found in Africa, and the California red scale, which is, of course, found in California. The interactions of these populations with their respective parasitoid populations were modeled using Lotka-Volterra models, and some interesting conclusions were reached for both systems:
  1. Very little evidence of aggregrated parasitoid attack.
  2. Age structure, in some form, is essential in order to correctly model host-parasitoid interactions.
  3. Local dynamics of the interactions are strongly influenced by parasitism refuges, in the form of physical refuges or host quality effects.
  4. Temporal refuges (i.e. syncroization issues) were not apparent, and host feeding did not significantly change overall dynamics.


Return to student paper summaries

The Dynamics of Arthropod Predator-Prey Systems

Project Categorization

(a) Mathematics Used:

This project uses two different types of equations to model the process as either continuous time or discrete time. To model the population of both the parasitoid and host in continuous time ordinary differential equations are used. To model the two populations in discrete time difference equations are used.

(b) Type of Model:

Various different kinds of population models are used. Most of them are just the Nicholson-Bailey and Lotka-Volterra models with add-ons and changes.

(c) Biological System Studied

The interaction between a host and parasitoid is studied. Specifically, this is of value to understand because of its application in farming where by one can introduce a natural pest killer (parasitoid) in instead of using chemical pesticides.

Citation of Paper

M. Rafikov, J.M. Balthazar, H.F. von Bremen, Mathematical modeling and control of population systems: Applications in biological pest control, Applied Mathematics and Computation, Volume 200, Issue 2, Special Issue on The Foz2006 Congress of Mathematics and its Applications, 1 July 2008, Pages 557-573, ISSN 0096-3003, DOI: 10.1016/j.amc.2007.11.036. (

This paper is a little less than ten years newer than the paper it cites. In this paper, they cite the paper that this wiki summarizes in order to show an example of an older way of thinking with a simpler model. They argue that only considering two species, the host and the parasitoid, is overly simple and other factors such as interactions between other species and the environment are important. In other words, the assumption that the system of interest is a closed one is invalid.

They take these new interactions into consideration by adding more differential equations to the model in the form of Lotka-Volterra equations. They then modify them as well.