Generalized stability conditions for host–parasitoid population dynamics: Implications for biological control

doi:10.1016/j.ecolmodel.2021.109656

Ecological Modelling

Volume 456, 15 September 2021, 109656

https://doi.org/10.1016/j.ecolmodel.2021.109656 Get rights and content

Highlights

•
Two orthogonal mechanisms stabilize a general class of host-parasitoid models.
•
Variation of risk causes host equilibrium density to increase with host reproduction.
•
Type III response causes host equilibrium to decrease with increasing reproduction.
•
Compared to Type I, a Type III response is more efficient for suppressing the host.

Abstract

Discrete-time models are the traditional approach for capturing population dynamics of insects living in the temperate regions of the world. We revisit classical discrete-time models of host–parasitoid population dynamics and provide novel results on the stability of the population dynamics. Discrete-time host–parasitoid models are characterized by update functions that connect the population densities from one year to the next, and a host escape response — the fraction of hosts escaping parasitism each year. For a general class of models we show that the stability can be simply characterized in terms of two quantities: the rate at which the host equilibrium changes with the host’s growth rate, and the sensitivity of the host’s escape response to the host density. Interestingly, stability is more likely to arise when the escape response is a decreasing function of the host density rather than an increasing function. Moreover, if the host’s escape response only depends on the parasitoid population density then the stability condition is further simplified to the host equilibrium density being an increasing function of the host’s reproduction rate. We interpret several mechanisms known for stabilizing host–parasitoid population dynamics in the context of these generalized stability conditions. Next, we introduce a hybrid approach for obtaining the update functions by solving ordinary differential equations that mechanistically capture the ecological interactions between the host and the parasitoid. This hybrid approach is used to study the suppression of host density by a parasitoid. Our analysis shows that when the parasitoid attacks the host at a constant rate, then the host density cannot be suppressed beyond a certain point without making the population dynamics unstable. In contrast, when the parasitoid’s attack rate increases with increasing host density (Type III functional response), then the host population density can be suppressed to arbitrarily low levels while maintaining system stability. These results have important implications for biological control where parasitoids are introduced to eliminate a pest that is the host species for the parasitoid.

Introduction

Insect population dynamics has been extensively studied using two different approaches: continuous-time and discrete-time models. The continuous-time framework is generally used to model populations with overlapping generations and all year-round reproduction. In contrast, discrete-time models are more suited for populations in temperate regions of the world that have non-overlapping generations and reproduce in a discrete pulse determined by season (Murdoch et al., 2003).

Here we revisit discrete-time models capturing the antagonistic interaction between two insect species (a host and a parasitoid). A typical life cycle of the host and the parasitoid is shown in Fig. 1, and consists of adult hosts emerging during spring, laying eggs that hatch into larvae. Host larvae then overwinter in the pupal stage, and metamorphosize as adults the following year. As adult hosts die after laying eggs, there is no overlap between generations across years. The host becomes vulnerable to attacks by a parasitoid wasp at one stage of its life cycle. For the sake of convenience, we assume the host’s vulnerable stage to be the larval stage. Adult female parasitoids emerge during spring, search and attack hosts by laying an egg into the body of the host. While adult parasitoids die after this time window, the parasitoid egg hatches into a juvenile parasitoid that grows at the host’s expense by using it as a food source, and this ultimately results in the death of the host. The juvenile parasitoids pupate, overwinter, and emerge as adult parasitoids the following year. There are more than 65,000 different species of parasitoid wasps and we refer the reader to Hajek, 2004, Godfray, 1994 and Waage and Greathead (1986) for fascinating details on parasitoid classification, life history and behavior. Synchronized life cycles, with no overlap of generations in both the host and the parasitoid makes discrete-time models highly appropriate for these systems.

We provide novel results on the stability of a general class of discrete-time models describing host–parasitoid interactions. Following a rich tradition of using hybrid or semi-discrete approaches to model ecological population dynamics (Pachepsky et al., 2008, Eskola and Geritz, 2007, Dugaw et al., 2004, Bonsall and Hassell, 1999, Briggs and Godfray, 1996, Geritz and Kisdi, 2004), we discuss a hybrid formulation that solves a nonlinear differential equation to obtain discrete-time models for host–parasitoid systems. Importantly, this mechanistic derivation of models using the hybrid approach can provide qualitatively different results compared to earlier phenomenological approaches. Finally, in the context of biological control of pests, we investigate different forms of parasitoid attack rates that lead to efficient suppression of the host population.

Section snippets

Host–parasitoid model formulation

A model describing host–parasitoid dynamics in discrete-time is given by $H_{t + 1} = R H_{t} f (R H_{t}, P_{t})$ $P_{t + 1} = k R H_{t} [1 - f (R H_{t}, P_{t})]$ where $H_{t}$ and $P_{t}$ are the adult host, and the adult parasitoid densities, respectively, at the start of year $t$ (Hassell, 2000a, Gurney and Nisbet, 1998, Singh et al., 2009, Marcinko and Kot, 2020, Schreiber, 2006, Livadiotis et al., 2015). If the host is vulnerable to the parasitoid at its larval stage, then $R H_{t}$ is the host larval density exposed to parasitoid attacks at the start of

General stability analysis

The fixed point of the discrete-time model (1) are the population densities that remain constant across years. Substituting $P_{t + 1} = P_{t} = P^{*}$ and $H_{t + 1} = H_{t} = H^{*}$ in (1) shows that apart from the trivial fixed point $P^{*} = H^{*} = 0$ , the non-trivial equilibrium is the solution to the following equations $1 = R f (R H^{*}, P^{*}), P^{*} = k (R - 1) H^{*},$ where $H^{*}$ and $P^{*}$ denote the host and parasitoid densities at equilibrium, respectively. For small perturbations close to $P^{*} = H^{*} = 0$ , the host dynamics in (1a) reduces to (3) resulting in an

Hybrid formulation of update functions

For a majority of host–parasitoid models the escape response $f$ in (1) is phenomenologically chosen or designed to recapitulate field observations. Recent work has proposed a mechanistic hybrid framework for deriving the discrete-time model, where ordinary differential equations are used to track population densities within the host vulnerable period of a given year (Singh and Nisbet, 2008, Emerick and Singh, 2016, Singh and Nisbet, 2007, Singh and Emerick, 2020). The solution of the

Limits of host suppression by parasitoids

We next use the hybrid formulation to investigate the suppression of host population density by a parasitoid. As done in other models (Neubert and Kot, 1992, Hassell and Varley, 1969b, May et al., 1981), our previous work (Singh and Nisbet, 2007) considered a self-limitation in the hosts ability to grow (due to finite food resources or attacks by other nature enemies) by considering a host mortality rate $γ_{L} = c_{h} L (τ, t)$ in (23) that is proportional to the host density. For a constant parasitoid

Conclusion

In this contribution, we have focused on the ecological interaction between hosts and parasitoids that holds tremendous potential in biological control of pests (Abram et al., 2016, Jervis et al., 1996, Ueno, 1998, Reeve and Murdoch, 1985, Jang and Yu, 2012, Hassell and Varley, 1969a, Kaser et al., 2018, Bompard et al., 2013). We provide novel results on the stability of host–parasitoid interactions modeled as per (1), and characterized by an arbitrary escape response $f (R H_{t}, P_{t})$ , which to be

CRediT authorship contribution statement

Abhyudai Singh: Conceived the study, Performed the mathematical analysis, Wrote the manuscript. Brooks Emerick: Performed the mathematical analysis, Wrote the manuscript.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (51)

AbramP.K. et al.
Parasitoid-induced host egg abortion; an underappreciated component of biological control services provided by egg parasitoids
Biol. Control
(2016)
BriggsC.J. et al.
The dynamics of insect-pathogen interactions in seasonal environments
Theor. Popul. Biol.
(1996)
CobboldC.A. et al.
The impact of parasitoid emergence time on host-parastioid population dynamics
Theor. Popul. Biol.
(2009)
DugawC.J. et al.
Seasonally limited host supply generates microparasite population cycles
Bull. Math. Biol.
(2004)
EmerickB.K. et al.
The effects of host-feeding on stability of discrete-time host-parasitoid population dynamic models
Math. Biosci.
(2016)
EmerickB. et al.
Global redistribution and local migration in semi-discrete host–parasitoid population dynamic models
Math. Biosci.
(2020)
GeritzS.A. et al.
On the mechanistic underpinning of discrete-time population models with complex dynamics
J. Theoret. Biol.
(2004)
NeubertM.G. et al.
The subcritical collapse of predator populations in discrete-time predator-prey models
Math. Biosci.
(1992)
SinghA. et al.
Semi-discrete host-parasitoid models
J. Theoret. Biol.
(2007)
TaylorA.D.
Heterogeneity in host-parasitoid interactions: ‘aggregation of risk’ and the ‘ $C V^{2} > 1$ rule’
Trends Ecol. Evol.
(1993)

AdlerF.R.

Migration alone can produce persistence of host-parasitoid models

Amer. Nat.

(1993)

BompardA. et al.

Host-parasitoid dynamics and the success of biological control when parasitoids are prone to allele effects

PLoS One

(2013)

BonsallM.B. et al.

Parasitoid-mediated effects: apparent competition and the persistence of host-parasitoid assemblages

Res. Popul. Ecol.

(1999)

CominsH.N. et al.

The spatial dynamics of host-parasitoid systems

J. Anim. Ecol.

(1992)

CroninJ.T. et al.

Host-parasitoid spatial ecology: A plea for a landscape-level synthesis

Proc. Biol. Sci.

(2005)

ElaydiS.

An Introduction to Difference Equations

(1996)

EskolaT.M. et al.

On the mechanistic derivation of various discrete-time population models

Bull. Math. Biol.

(2007)

GodfrayH.C.J.

Parasitoids; Behavioral and Evolutionary Ecology

(1994)

GurneyW.S.C. et al.

Ecological Dynamics

(1998)

HajekA.E.

Insect parasitoids: attack by aliens

HassellM.P.

The Spatial and Temporal Dynamics of Host Parasitoid Interactions

(2000)

HassellM.

Host–parasitoid population dynamics

J. Anim. Ecol.

(2000)

HassellM.P. et al.

Sigmoid functional responses and population stability

Theor. Popul. Biol.

(1978)

HassellM.P. et al.

The persistence of host–parasitoid associations in patchy environments. I. A general criterion

Am. Nat.

(1991)

HassellM.P. et al.

New inductive population model for insect and its bearing on biological control

Nature

(1969)

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Generalized stability conditions for host–parasitoid population dynamics: Implications for biological control

Highlights

Abstract

Introduction

Section snippets

Host–parasitoid model formulation

General stability analysis

Hybrid formulation of update functions

Limits of host suppression by parasitoids

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

Biol. Control

Theor. Popul. Biol.

Theor. Popul. Biol.

Bull. Math. Biol.

Math. Biosci.

Math. Biosci.

J. Theoret. Biol.

Math. Biosci.

J. Theoret. Biol.

Trends Ecol. Evol.

Migration alone can produce persistence of host-parasitoid models

Amer. Nat.

Host-parasitoid dynamics and the success of biological control when parasitoids are prone to allele effects

PLoS One

Parasitoid-mediated effects: apparent competition and the persistence of host-parasitoid assemblages

Res. Popul. Ecol.

The spatial dynamics of host-parasitoid systems

J. Anim. Ecol.

Host-parasitoid spatial ecology: A plea for a landscape-level synthesis

Proc. Biol. Sci.

An Introduction to Difference Equations

On the mechanistic derivation of various discrete-time population models

Bull. Math. Biol.

Parasitoids; Behavioral and Evolutionary Ecology

Ecological Dynamics

Insect parasitoids: attack by aliens

The Spatial and Temporal Dynamics of Host Parasitoid Interactions

Host–parasitoid population dynamics

J. Anim. Ecol.

Sigmoid functional responses and population stability

Theor. Popul. Biol.

The persistence of host–parasitoid associations in patchy environments. I. A general criterion

Am. Nat.

New inductive population model for insect and its bearing on biological control

Nature