A Model For Sea Lice (Lepeophtheirus salmonis) Dynamics In A Seasonally Changing Environment

Sea lice (Lepeophtheirus salmonis) are a significant source of monetary losses on salmon farms. Sea lice exhibit temperature-dependent development rates and salinity-dependent mortality, but to date no deterministic models have incorporated these seasonally varying factors. To understand how environmental variation and life history characteristics affect sea lice abundance, we derive a delay differential equation model and parameterize the model with environmental data from British Columbia and southern Newfoundland. We calculate the lifetime reproductive output for female sea lice maturing to adulthood at different times of the year and find differences in the timing of peak reproduction between the two regions. Using a sensitivity analysis, we find that sea lice abundance is more sensitive to variation in mean annual water temperature and mean annual salinity than to variation in life history parameters. Our results suggest that effective sea lice management requires consideration of site-specific temperature and salinity patterns and, in particular, that the optimal timing of production cycles and sea lice treatments might vary between regions.

1 Introduction vironmental trends, as well as predicting the effectiveness of different treatment regimes. 23 These two needs can be accomplished through mathematical modelling and it is imperative 24 that tractable and biologically sound models are developed to aid practitioners in decisions 25 regarding sea lice dynamics. 26 Seasonal environmental variability plays a major role in the dynamics of many disease  Table 1), thus models of sea lice dynamics must be able 29 to incorporate the effects of seasonally varying temperature and salinity on the sea louse   The length of time that a nauplius or chalimus requires to mature to their respective 72 next life stages depends on water temperature (Table 1). Let γ x (T (t)) be a function that 73 describes the rate of change in the level of development for a given stage x ∈ {P, C} as it 74 depends on temperature (T ), which changes over time (t). For notational simplicity, we write 75 simply γ x (t), because given functions that describe how temperature changes with respect to 76 time (T (t)), and how the development rate changes with respect to temperature (γ x (T )), we 77 can then determine how the development rate changes with respect to time (γ x (t)) without 78 needing to explicitly reference the dependence on temperature. 79 The waiting times associated with maturation are such that a cohort exiting a state x at 80 time t, will all have entered that stage at t − τ x (t). The waiting time, τ x (t), depends on the 81 development rate, γ x (t), and is defined as the length of time that it takes sea lice to reach 82 a threshold development level,q x , given that they entered the stage x with a development (1) (Nisbet & Gurney, 1983).

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Natural mortality occurs in all stages at a per capita rate µ y (S(t)), where y ∈ {P, I, C, A}.

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Natural mortality is a function of salinity S(t), which is a function of time (t). For notational 87 simplicity, we write simply µ y (t), because given functions that describe how salinity changes 88 with respect to time (S(t)), and how the mortality rate changes with respect to salinity 89 (µ y (S)), we can then determine how the mortality rate changes with respect to time (µ y (t)) 90 without needing to explicitly reference the dependence on salinity. Not all members of a 91 cohort who enter a stage x at time t − τ x (t) survive to mature at time t. The proportion of 92 the cohort that survive the maturation period is, The proportion of eggs that produce viable nauplii is a function of salinity, v(t). All other 94 events in the sea lice life history do not depend on temporally varying quantities and are 95 assumed to occur at constant per capita rates. The complete model is a system of delay 96 differential equations, where η is the number of eggs per egg string, is the rate of egg string production, ι is the 98 rate of infection per fish, f is the number of fish on the farm, and all model parameters are 99 summarized in Table 2. Equations (6) and (7) arise from differentiating equation (1) with 100 respect to time (Nisbet and Gurney, 1983). The γ x (t)/γ x (t − τ x ) terms arise because we 101 wanted to ensure a correspondence between our model (which lumps all individuals with a 102 development level q x <q x together into one state) and a model that treats the development 103 level as a continuous quantity (Nisbet & Gurney, 1983; see Appendix A for further details).

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The shape of the function is described by the duration of the life stage (β −2 x ) at the reference Water temperature (T (t)) and salinity (S(t)) on salmon farms varies over time. We use 121 sinusoidal functions to describe the general annual patterns, where a is the average annual temperature, c is the average annual salinity, and b and d are 123 the respective amplitudes of the cosine functions. Sinusoidal functions of the form, were fit to monthly temperature and salinity data from a salmon farm in the Broughton

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Mortality is related to salinity via linear and log-linear relationships from the literature.

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The salinity-mortality relationship (µ A (S)) is log-linear for adult sea lice (Connors et al.

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2008). We assume that the mortality rate for adults and chalimi (µ C (S)) is similar. The (2006), which is fit to data from Johnson & Albright (1991). We assume that the mortality 134 rate for nauplii and copepodids (µ I (S)) is similar.

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The number of eggs per egg string (η) was parameterized using data from Heuch et al.  lower salinities in warmer climates (Fig. 3). Environmental conditions that result in < 1 are shown as circles.
The basic reproductive ratio, R 0 , is commonly used as a measure of reproductive success and acts as a threshold condition that indicates either population persistence or extinction 166 (Caswell 2009). When R 0 ≥ 1 the population will grow with each subsequent generation and 167 persist, whereas when R 0 ≤ 1 the population will shrink with each subsequent generation 168 until extinction. In seasonal systems, R 0 will depend on the time that the infection is 169 introduced to the system. We define R 0 (t) such that it is the number of second generation  salinities and visa versa. We find R 0 (t) to be highest in December, when salinity is high, but 182 temperatures are low (Fig. 4A). As such, sea lice that enter the farm in December will go 183 on to produce the most offspring despite having longer generation times than sea lice that 184 hatch in the summer months (Fig. 4B). The value of R 0 (t) is not < 1 at any point during   The simulation begins with only adult females and the only parameter that affects adult 208 mortality is mean salinity (c, Fig. 5A). After the cohorts start maturing the size of the 209 adult female population is also affected by parameters relating to maturation, infection, and 210 reproduction ( Fig. 5A and B). The three most sensitive parameters at 180 days were mean 211 temperature (a), mean salinity (c), and the number of eggs per egg clutch (η; Fig. 5A and B).

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There is a substantial difference between the timing of the peak in R 0 (t) for British 227 Columbia and Newfoundland. We found that the peak value of R 0 (t) for both the British

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Columbia and the Newfoundland sites occurred during peak salinity levels, although in

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Newfoundland the salinity levels were fairly constant and the highest salinity levels also 230 coincided with the highest sea surface temperatures. As such, optimal treatment schemes will 231 also differ between these two sites. In both the Broughton Archipelago in British Columbia data would need to be provided for analysis using our model. This is especially pertinent 251 as salinity patterns may vary substantially over small spatial scales due to their proximity 252 to rivers, and even two sites within the same broad geographic region potentially could have 253 very different salinity patterns.

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It is important to note that the R 0 (t) we calculate is not a threshold condition for sea lice 255 epidemics, since subsequent generations will hatch throughout the year and experience their sea lice outbreaks. One of the advantages to using a deterministic delay differential equation 261 approach is that the theoretical approaches outlined in Zhao (2015) can be utilized.

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Our sensitivity analysis found that adult female sea lice abundance is most sensitive to 263 average annual temperature and salinity. This is likely because a large number of parameters 264 depend on temperature (τ P (t) and τ C (t)), salinity (µ P (t), µ I (t), µ C (t), and µ A (t)) or both 265 (φ P (t), φ C (t)). Our findings that lice abundance is more sensitive to the development rate of