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Fish culture
Fertilized eggs of chum salmon [Oncorhynchus keta (Walbaum, 1792)] were obtained from Unosumai hatchery, Iwate, Japan. The eggs were fertilized artificially on 4th December 2018 and eye-stage embryos were transferred to the Atmosphere and Ocean Research Institute, Chiba, Japan. Hatched salmon fries were fed with commercial diet after emergence in freshwater. Natural seawater (SW, salinity 35 ‰) was obtained from the Kuroshio Current at Hachijō-jima. The SW was stored in underground facilities and salmon eDNA was not detectable from the SW stock from in-house experiment. Salmon juveniles were transferred to seawater tanks after 2-months culture in freshwater. The experimental tank contained 500 L seawater with recirculation and temperature control at 14 °C. Twenty-five individuals (fork-length: ca. 20 cm; weight: ca. 50 g) were kept in the tank and the water served as a source of eDNA in the present study. All animal studies were performed according to the Guideline for Care and Use of Animals approved by the Animal Experiment Committee of The University of Tokyo.
Filtration of water
Two concentrations of salmon water were prepared. “Neat salmon water” was obtained directly from the stock tank. “Diluted salmon water” was prepared by diluting the tank water 10 times with seawater that had not been used for any previous fish culture. To standardize filtration conditions, 1 L of salmon water at either concentration was filtered through a Sterivex cartridge (0.45 μm, Millipore SVHV010RS, Merck, Tokyo, Japan) using a vacuum manifold (Manifold KMP-3, Advantec, Tokyo, Japan) attached to an aspirator (Eyela A-1000S, Tokyo Rikakikai, Tokyo, Japan) (Fig. 7). After filtration, the outlet of the Sterivex cartridge was sealed by Parafilm and 1.6 mL of RNAlater was introduced into the cartridge through the inlet. The inlet was subsequently sealed by Parafilm and the cartridge was stored at − 20 °C until further processing. All the Sterivex cartridges were then randomly assigned to different extraction protocols after the filtration step to reduce bias on filtration order.
Figure 7

Representative photos of Sterivex cartridges with different extraction protocols. (A) A cartridge extracted by 0.44 mL lysis mix protocol. Note the water mark that is present on the filter surface (white arrow), suggesting insufficient contact with the lysis buffer mix. (B) A cartridge extracted by 2.0 mL lysis buffer mix with backwash. Note that no water mark is found as in (A). (C) A cartridge centrifuged by angled rotor. Note that remaining lysate is present (black arrow).

Full size image

DNA extraction and purification
All the connectors, stoppers, and silicon tubes were decontaminated by bleaching before use5. The RNAlater in the Sterivex cartridges was thawed on ice and removed by aspiration through the outlet. A silicon tube (1.5 cm; i.d. 3 mm; o.d. 9 mm) was used to connect a Luer-Lock adaptor (VRF306, AS ONE, Osaka, Japan) to the outlet such that both inlet and outlet can be sealed by Luer-Lock stoppers (VRMP6, AS ONE, Osaka, Japan). We tested three protocols that were modified from a published method that was widely used in the eDNA field13. The following reagents are based on the Qiagen DNeasy Blood and Tissue Kit (ThermoFisher Scientific, Waltham, MA, USA), with additional reagents (e.g. Buffer AL and proteinase K) purchased from the same sources when necessary. The major modifications are summarized in Fig. 7. For Protocol 1, we followed the original extraction protocol13. A lysis buffer mix (PBS 220 μL, Buffer AL 200 μL, Proteinase K 20 μL; 440 μL total volume) was introduced to the cartridge through the inlet. Both ends of the cartridge were sealed by Luer-Lock stoppers and the cartridge was incubated at 56 °C for 30 min with mild rotation. In Protocol 2, approximately four-times volume (2 mL) of lysis buffer mix (PBS 990 μL, Buffer AL 910 μL, Proteinase K 100 μL) was introduced through the inlet. After sealing the ends, the cartridge was incubated at 56 °C for 30 min without rotation. The rotation was omitted because the filter surface was completely covered by the lysis buffer mix. In Protocol 3, 2 mL lysis buffer mix (PBS 990 μL, Buffer AL 910 μL, Proteinase K 100 μL) was introduced into the cartridge through the outlet using a 2.5 mL syringe (Terumo Corporation, Tokyo, Japan) to flush the cartridge in a reverse direction. After sealing the ends, the cartridge was incubated at 56 °C for 30 min without rotation.
After the incubation, each Sterivex cartridge was placed in a spin column (maxi spin, flat bottom, Ciro Manufacturing Corporation, Deerfield Beach, FL, USA) attached to a 50 mL centrifuge tube, with the cartridge inlet facing downward. The unit was centrifuged with a swing-type rotor at 2500g for 10 min at 25 °C to elute the content. The cartridge and spin column were removed and molecular grade ethanol (99.5%, Fujifilm Wako Pure Chemical Corporation, Osaka, Japan) was added to one-third of the final volume (200 μL for Protocol 1 and 1 mL for Protocols 2 and 3). The mixture was loaded on a DNeasy Blood and Tissue Kit column attached to a vacuum manifold and the column was washed by 0.8 mL AW1 buffer and 0.8 mL AW2 buffer sequentially. The DNeasy column was dried by centrifugation at 17,700g for 2 min. Adsorbed DNA was eluted from the column with 75 μL AE buffer twice to maximize recovery. A total of 150 μL sample was collected from each extraction. Six replicates of Sterivex cartridges were used for each protocol. To test the extraction efficiency, each Sterivex cartridge was extracted 3 times and the eluted samples were analyzed separately.
An aliquot of DNA sample (100 μL) was further purified by Qiagen DNeasy PowerClean Pro Cleanup Kit (ThermoFisher Scientific, Waltham, MA, USA) according to the manufacturer’s protocol.
To test the binding capacity of the DNeasy column, we performed an additional experiment to determine whether the quantity of sample input will have a saturating effect on the column binding. We prepared a high sample input group by filtering 2 L of neat salmon seawater per Sterivex cartridge, while 1 L of 10×-diluted salmon seawater per cartridge was considered as a low sample input group. The extraction was performed using Protocol 3. Since we hypothesized that the binding capacity of one DNeasy column could be saturated by high input, we compared the eDNA extracted by a single column or double columns connected in series. In the case of double columns, the elution was performed individually for the upper and lower columns, and the eluates were analyzed separately.
Chum salmon eDNA quantification
The eDNA was measured by quantitative PCR (qPCR) as described in a previous study5, with slight modification. Instead of using ABI TaqMan Environmental Master Mix 2.0 (Applied Biosystems, Foster City, USA), Takara Probe qPCR Mix (Takara Bio Incorporation, Kusatsu, Japan) was used as we showed that this reagent has a higher capacity to resist inhibition by environmental contaminants in the PCR reaction (see Results). We compared the resistance to contaminants by Environmental Master Mix and Takara Probe qPCR Mix according to the spiking/dilution methods described previously5, using the environmental samples collected from Murohama Bay of Otsuchi, which contained high levels of PCR inhibitors.
Method validations in field samples
Thirty-four random seawater samples (1 L each) were collected from the Otsuchi River mouth (141′26.917 E, 39′55.021 N), Iwate Prefecture between March 15th–19th, 2020. Negative control was prepared by filtering distilled water (1 L). The water samples were filtered and treated with RNAlater as described in previous section. The cartridges were extracted according to the Protocol 3 described in previous section. To test the extraction and purification efficiencies, 1010 copies of a control plasmid (self-ligated pGEMT-easy, Promega, Madison, WI, USA) were added to each 2 mL lysis buffer mix (PBS 990 μL, Buffer AL 910 μL, Proteinase K 100 μL) at the beginning of the extraction. The extraction efficiency was calculated from ratio of control plasmid quantified from the DNeasy-extracted samples relative to input quantity. Real-time PCR for control plasmid was designed at the M13 regions: forward primer (pGEMT-F), 5′-TTTCCCAGTCACGACGTT-3′; reverse primer (pGEMT-R), 5′-TTCACACAGGAAACAGCTATGA-3′; probe (pGEMT-probe), 56/FAM/ACGCGTTGG/ZEN/ATGCATAGCTTGAGTA/3lABkFQ (Integrated DNA Technologies Inc., Coralville, IO, USA). Chum salmon eDNA concentrations of both DNeasy extracted samples and samples further purified by PowerClean Pro Cleanup Kit were measured to estimate the recovery rate of PowerClean purification. To estimate the PCR inhibitor effects from samples without purification by PowerClean Pro Cleanup Kit, an additional assay was performed with 105 copies of chum salmon standard spiked into each PCR reaction. The chum salmon standard used in spiking is the same plasmid DNA5 used for constructing the standard curve. A reduction in spiked values calculated by subtracting the spiked copy number to original copy number indicates that PCR reaction was not optimal.

$$ {text{Detection}},{text{rate}},{text{of}},{text{spiked}},{text{DNA}} = frac{{left( {{text{copy}},{text{in}},{text{spiked}},{text{sample}} – {text{copy}},{text{in}},{text{original}},{text{sample}}} right)}}{{left( {10^{5} ,{text{copy}},{text{of}},{text{spiked}},{text{DNA}}} right)}} times 100% $$

Detection rates lower than 100% indicate that the PCR reaction could be affected by PCR inhibitors in environmental samples.
To test whether addition of bovine serum albumin (BSA) can further protect the real-time PCR reaction from non-specific inhibition, a final concentration of 1 μg/μL BSA (A4161, Sigma-Aldrich, St. Louis, MO, USA) was added to each Takara Probe qPCR Mix reaction. Real-time PCR was performed on the serially-diluted inhibitor-rich sample and the slope of the dilution curve was determined. PCR efficiencies were calculated according to the following equation (Efficiency = 10−1/slope)27. As we found improved quantification from BSA experiment, we further tested chemical additives including 2% DMSO (D2650-5X Sigma-Aldrich, St. Louis, MO, USA), 0.5% Tween 20 (P9416 Sigma-Aldrich, St. Louis, MO, USA), 0.01% Formamide (F9037 Sigma-Aldrich, St. Louis, MO, USA), and a protein-based additive GP-32 protein (Nippon Gene, Toyama, Japan) at 0.1 μg/μL final concentration28. A pooled eDNA sample was made from mixing some inhibitor-rich samples from the 34 random environmental samples, and subsequently used in this preliminary test. The PCR amplification plots were compared between chum salmon standard and the pooled sample. Slope of the PCR amplification plot was obtained by linear regression on the steepest linear portion of the curve. PCR reaction was compromised by the inhibitor when we observed a reduction in saturated fluorescence and a flatten slope of the linear portion of the amplification plot of the pooled sample in comparison to those of standard. We considered that the rescue from reduction in saturated fluorescence and slope change are indicating an increase in resistance to PCR inhibitors by the additive.
Statistical analysis
The concentrations of salmon eDNA extracted by various protocols were analyzed by two-way ANOVA followed by Tukey’s multiple comparisons (GraphPad Prism Ver. 6 for Windows, San Diego, CA, USA). Statistical significance (p  More

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On a summer morning in 2019, Andy Suhrbier pilots a small aluminium boat out to a mussel raft in a quiet cove on the eastern shore of Puget Sound in Washington State. As the boat approaches, a mother seal and her pup resting on the raft slip into the water. Suhrbier climbs from his boat onto the raft; the only sign of life is a vague smell.
Suhrbier tugs on a couple of ropes attached to one of the raft’s beams. Soon, a mesh-lined plastic cage emerges with water and silt pouring out of it. He picks off several sea stars and tosses them back into the water, then flips open the lid like a pirate opening a treasure chest.
Inside is more dark sediment — mostly waste from the mussels, the source of the smell. Suhrbier sifts through it. He is looking for something.
“Look at this monster!” he says, holding up a sea cucumber nearly a foot long. Its deep red body covered in orange bumps stands out from the muck like a gold doubloon. “That’s definitely market size.”

Suhrbier is a biologist with the Pacific Shellfish Institute in Olympia, Washington, a non-profit research organization that works to promote healthy wild shellfish populations and sustainable shellfish aquaculture along the US west coast. Two years earlier, he had put sea cucumbers in cages and suspended them beneath the mussel raft, as part of an effort to develop aquaculture in Puget Sound. The hefty size of the cucumbers is a promising sign.
Suhrbier and his colleagues think that sea-cucumber farming could have two benefits. First, the animals could help to prevent excess waste from building up underneath aquaculture installations, such as mussel rafts or net pens used to hold bony fish such as salmon. (Sea cucumbers, soft-bodied animals related to sea urchins, move slowly over the sea floor eating detritus — the vacuum cleaners of the ocean.) Second, a ready source of farmed sea cucumbers could reduce the poaching of wild stocks to feed the growing market in east and southeast Asia.
Globally, aquaculture produced 82.1 million tonnes of aquatic animals in 2018, and wild fisheries produced 97 million tonnes, according to the United Nations’ Food and Agriculture Organization (FAO). But the value of farmed fish was higher, around US$250 billion compared with $151 billion for wild-caught fish. Aquaculture production of animals is projected to increase by one-third by 2030, reaching 109 million tonnes, and will supply the majority of aquatic protein in people’s diets by 2050.
“We need to grow the amount of seafood available, as world populations grow, to provide enough protein for everybody,” says Monica Jain, founder of Fish2.0 in Carmel, California, an organization that promotes investment in sustainable seafood businesses. With the catches from wild fisheries remaining largely flat and some stocks already overexploited, “aquaculture is really the only way to do that”. But as the industry grows, Jain and other aquaculture advocates want to make sure that it does so sustainably.
Double alchemy
Aquaculture is a relatively small proportion of the global food system — terrestrial meat production (both livestock and wild game) totalled around 342 million tonnes in 2018, and production of grains and cereals was 2.7 billion tonnes. However, aquaculture is more diverse, particularly in terms of the animals farmed. These range widely across taxonomic groups, including bony fish (carp, tilapia and salmon, for example), crustaceans (shrimp, prawns and crayfish), molluscs (clams, oysters and mussels) and echinoderms (sea cucumbers). Various species of seaweed are also gathered. There are freshwater, saltwater, brackish water and self-contained terrestrial aquaculture systems. And each has its own sustainability benefits and challenges.
One subsector that offers huge environmental advantages and has no equivalent in terrestrial agriculture is non-fed aquaculture. Marine bivalves, such as clams, mussels and oysters, get their nourishment by filtering microscopic plants, detritus and nutrients from the water that surrounds them. They require minimal inputs and can even improve the water quality. In this sense, Suhrbier’s sea cucumbers represent a kind of double alchemy: non-fed aquaculture species grown on the wastes of other non-fed aquaculture species.
Similarly, cultivated seaweeds can remove excess nutrients, such as nitrogen, that contribute to the formation of areas of oxygen-poor water where marine life has difficulty surviving, known as dead zones. By taking up carbon they can also help to alleviate ocean acidification at a local scale. Moreover, “seaweeds are incredibly nutritious,” says Alecia Bellgrove, head of the DeakinSeaweed Research Group at Deakin University in Melbourne, Australia. “They are, for example, fantastic sources of trace minerals, which are often lacking in our diets based on terrestrial foods.”

Sea cucumbers are retrieved from the mesh-lined cages at Puget Sound in Washington state. Credit: Sarah DeWeerdt

Aquatic animals that require feed — mainly prawns and bony fish — also have an environmental advantage over animals raised in terrestrial agriculture. Because most are cold-blooded, they convert food into body mass more efficiently than birds and mammals, which need energy to help regulate their body temperature. So it takes less feed to produce a kilogram of salmon, for example, than it does to produce a kilogram of, say, beef or pork.
However, some of the most lucrative aquaculture species are carnivorous, and therefore sit higher in the food chain than any terrestrial species raised in agriculture. Take the Atlantic salmon (Salmo salar), for example. In the mid-2000s, salmon aquaculture, now a $15.4-billion industry, was growing rapidly. Feeding the salmon demanded an increasing share of the world’s fish meal and fish oil, which was sourced from small forage fish, such as anchovies, sardines and capelin. But while demand from the salmon farms grew, fishing yield for the forage species remained relatively flat. It took at least 4 kilograms of wild-caught forage fish to produce just 1 kilogram of salmon.
From an environmental point of view, “It made no sense,” says Scott Nichols, founder of Food’s Future, a consultancy in West Chester, Pennsylvannia, which promotes the development of sustainable aquaculture businesses. As a biochemist working at US chemical company DuPont in the mid 2000s, Nichols helped to develop a way to produce omega-3 fatty acids from yeast. The fatty acids were then incorporated into salmon feed to replace some of the wild-fish component. The new feed was tested through a partnership between DuPont and the aquaculture firm AquaChile based in Puerto Montt, Chile, in the form of the salmon producer Verlasso in Miami, Florida.
“We were able, after a couple of years of production, to get to the point where for every kilo of salmon that was produced, we were using less than a kilo of wild-caught fish,” Nichols says. “So our farming practices resulted in the net production of fish on the planet.”
Other companies soon joined in, producing omega-3s in genetically engineered canola oil or single-celled algae. Meanwhile, fish-oil and fish-meal producers are increasingly making use of fish trimmings and other by-products that previously went to waste. Fish meal and fish oil, which are still used in a variety of aquaculture feeds, as well as in products such as food supplements, accounted for around 10% of the world’s total fish production in 2018, according to the FAO. But nonetheless, Nichols takes heart from developments. “What looked on the face of it to be dismal in 2006 now looks to be very promising,” he says.
Disease detectives
An increasingly important threat to aquaculture sustainability is disease, which affects all subsectors of aquaculture and causes an estimated $6 billion worth of aquatic animal losses every year. Diseases include parasites called sea lice in salmon; white spot syndrome virus in prawns, which emerged in the early 1990s and devastated prawn farming throughout Asia before spreading to the Americas; and tilapia lake virus, which threatens the economic and nutritional gains that freshwater aquaculture has made possible in many low- and middle-income countries.
As aquaculture is scaled up, the problem of disease will also become greater. “As you expand the volume of production, you are going to get significant losses,” says Grant Stentiford, a pathologist and head of aquatic animal health at the Centre for Environment Fisheries and Aquaculture Science, Weymouth, UK. “You’ve used up potentially large amounts of resource to get absolutely nowhere.”
To deal with such threats, some large producers who supply the export market are moving to self-contained, land-based systems. Others are moving away from the coast into deeper waters that might dilute the threat of disease. Vaccines have also made a difference, reducing not only the threat of many fish diseases, but also antibiotic use — another major environmental concern about the industry. And high-throughput sequencing of the microbial DNA in aquaculture systems could provide early warning of disease outbreaks.
But many of these solutions are expensive and, therefore, out of reach for the small and medium-sized producers who make up the majority of the global aquaculture industry, producing food for subsistence or local markets in low- and middle-income countries. Moreover, diseases that threaten aquaculture are emerging every three to five years on average. The dearth of knowledge about aquatic pathogens makes diseases hard to predict and spot.
It can also be a challenge to deduce their cause. For example, ice-ice disease results in bleaching of Kappaphycus seaweed, which is grown in large amounts in southeast Asia and Tanzania for the production of food additives, such as the thickening agent carrageenan. The disease has caused yields to plummet over the past decade, but “the causative agent is still not known”, says Valéria Montalescot, senior project manager for GlobalSeaweedSTAR, a four-year research project based at the Scottish Association for Marine Science in Oban, UK, which aims to boost knowledge about seaweed cultivation in low- and middle-income countries. Kappaphycus is usually grown from cuttings, so the whole crop across multiple countries might be the result of just a few clones, possibly making it more vulnerable to disease, Montalescot adds.
Diverse yields
Climate change is complicating efforts to fight disease. Higher water temperatures can alter the microbial community of a body of water, encouraging the growth of pathogens, as well as stressing organisms and making them more vulnerable to disease. One suggested cause of ice-ice disease is that temperature-stressed seaweeds release compounds that attract bacteria, for example.
And temperature is not the only issue. Both increased rainfall and salinity intrusion from sea-level rise can alter water chemistry in ways that are detrimental to aquaculture organisms. Storms can destroy aquaculture crops or infrastructure in the water and on land. “From an environmental point of view I think climate change is the greatest challenge” for the sustainability of aquaculture systems, says Nesar Ahmed, who studies global seafood sustainability at Deakin University.
Climate change also intersects with aquaculture’s pressure on water and land resources. Inland aquaculture demands 429 cubic kilometres of fresh water each year — much less than the demand from terrestrial agriculture, but still enough to pose a strain on increasingly drought-prone areas.
In south and southeast Asia, prawn cultivation has contributed to the destruction of 38% of the world’s mangrove habitats, which have a variety of important ecological functions, including sequestering carbon and buffering coastlines from storms and sea-level rise. The loss of mangroves has also resulted in saltwater intrusion rendering inland areas unsuitable for terrestrial agriculture.
Some farmers are now producing prawns among intact mangrove stands. Although there are concerns that this practice might also damage the health of the mangroves, it is part of a larger trend to create aquaculture systems that include multiple species and involve interrelationships more like the ones that keep natural ecosystems in balance.
Some examples of this integrated aquaculture are long-established, such as stocking rice paddy fields with fish or prawns. The animals eat pests and fertilize the rice crop, increasing rice yields and providing an extra source of protein or income for small-scale farmers, Ahmed says. Growing two species in a single body of water also reduces overall water use.
This type of rice–fish system has been practised for hundreds of years in China and has been designated a Globally Important Agricultural Heritage System by the FAO — a designation that aims to preserve agricultural knowledge that can contribute to a more sustainable and resilient food system. Large-scale aquaculture operations, such as Cooke Aquaculture based in Blacks Harbour, Canada, have also been experimenting with multi-species systems. The company keeps salmon in net pens near both mussel and kelp rafts in the Bay of Fundy, Canada.

In theory, integrated aquaculture can help to increase yields, decrease risk by diversifying operations, and is generally a more environmentally sound form of aquaculture. But in practice, it can be difficult to quantify these benefits. For example, because nitrogen moves freely through water, it is difficult to track uptake of excess nitrogen produced by bony fish by seaweed growing nearby. And then there are the complexities of managing an operation with multiple species — not just producing them but also harvesting, processing and marketing them.
Suhrbier knows such difficulties well. The sea cucumbers he and his team harvested from under the mussel raft were the right size, weight and colour for the export market, but the mussel producer he was working with was unable to renew its permit at that location. The raft was lost, and with it Suhrbier’s chance of follow-up experiments to develop sea-cucumber aquaculture techniques. “I was really shocked and saddened to see that go because it was one of those places where it just makes a lot of sense for sea cucumbers to be,” Suhrbier says. The new location of the producer’s rafts isn’t a good habitat for sea cucumbers.
Suhrbier is still experimenting growing sea cucumbers alongside other types of aquaculture operation around the Puget Sound area. But, like an increasing number of aquaculture researchers, he is beginning to think that producing the animals needs to move in a simpler and more radical direction. Growing sea cucumbers in cages is labour intensive. What if the animals are placed in the vicinity of aquaculture operations and left to roam freely — like a marine equivalent of a ranch or even a permaculture system?
“If we could mainly enhance the wild population around these areas, I think that would be a great benefit for everybody,” Suhrbier says. “I’m trying to have something that fits in: easy, cost effective and as passive as it can be.” More

The long-term population dynamics (i.e., extinction or persistence in an equilibrium value) of the system (2) is mainly dependent on the constant allocation of energy between life-history functions. In the case of population persistence, there is a unique long-term allocation strategy towards reproductive functions that maximizes the population abundance and minimizes individual consumption. In addition, this strategy is dependent on the parameters associated with both fertility ((kappa ) and ({mathscr {B}}_0)) and survival ((xi _{c}) and (e_{1/2})) costs.
Preliminary results
To investigate long-term dynamic patterns, we will relate the energy states of reproduction and maintenance, resource density and population abundance at the end of reproductive seasons, namely in the time sequence ({ntau ^+}_{nge 0}). In the system (4) the relationships between the state variables are the same for each (tau ) unit of time. Then, there is a transformation that relates to the vector ( (R,E_ {r}, E_ {m},P) ((n + 1)tau ^+) ) with ( (R,E_ {r}, E_ {m}, P) (n tau ^+) ) such that ( (n + 1) tau ^+ – ntau ^+ = tau ). This relationship is determined by the following discretization (or stroboscopic map) of the impulsive differential system (4):

$$begin{aligned} left{ begin{array}{lll} E_{r}((n+1)tau ^+)&=(1-kappa gamma )[E_{r}(ntau ^+)+alpha Phi (n,P(ntau ^+))],\ &{}&{}\ E_{m}((n+1)tau ^+)&=dfrac{{mathscr {B}}_0[E_{m}(ntau ^+)+(1-alpha )xi _{c}Phi (n,P(ntau ^+))]}{{mathscr {B}}_0+gamma [E_{r}(ntau ^+)+alpha Phi (n,P(ntau ^+))]},\ P((n+1)tau ^+)&=left{ 1-mu +dfrac{gamma [E_{r}(ntau ^+)+alpha Phi (n,P(ntau ^+))]E_{m}((n+1)tau ^+)}{e_{1/2}+E_{m}((n+1)tau ^+)}right} P(ntau ^+)e^{-lambda tau } end{array} right. end{aligned}$$
(6)

and (R(ntau ^+)=K), where the function (Phi ) defined by (5) is evaluated at (t=(n+1)tau ) and extended to (p=0):

$$begin{aligned} Phi (n,p)=left{ begin{array}{ll} dfrac{R_{max}(e^{lambda tau }-1)}{lambda }, &quad text {if}, p=0, \ &{}\ dfrac{K-R((n+1)tau )e^{lambda tau }}{p}+dfrac{lambda }{p}displaystyle int _{ntau }^{(n+1)tau }{R(s)e^{lambda (s-ntau )}ds}, & quad text {if}, pne 0, end{array} right. end{aligned}$$
(7)

for any (nge 0). Indeed, in non-reproductive seasons the consumer–resource dynamics are described by the continuous component of the system (4), which can be solved. Directly we have (P(t)=P(ntau ^+)e^{-lambda (t-ntau )}) for any (tin (ntau ,(n+1)tau ]). In addition, the per capita consumption rate (1) can be expressed in terms of ( R'(t)). Therefore, reproductive and maintenance energy rates are determined by

$$begin{aligned} E_r'(t)=-alpha frac{R'(t)e^{lambda (t-ntau )}}{P(ntau ^+)}quad text {and}quad E_m'(t)=-(1-alpha )xi _{c}frac{R'(t)e^{lambda (t-ntau )}}{P(ntau ^+)},quad tin (ntau ,(n+1)tau ]. end{aligned}$$

Integrating these functions in the interval ( (n tau , t] ), we obtain

$$begin{aligned} E_r(t)&= {} E_{r}(ntau ^+)+alpha Phi (t,R(t),P(ntau ^+)) end{aligned}$$
(8)

$$begin{aligned} E_m(t)&= {} E_m(ntau ^+)+(1-alpha )xi _{c}Phi (t,R(t),P(ntau ^+)). end{aligned}$$
(9)

Evaluating Eqs. (8)–(9) at ( t = (n + 1)tau ) (end of the non-reproductive season), we have

$$begin{aligned} E_r((n+1)tau )&= {} E_{r}(ntau ^+)+alpha Phi (n,P(ntau ^+)), end{aligned}$$
(10)

$$begin{aligned} E_m(n+1)tau )&= {} E_m(ntau ^+)+(1-alpha )xi _{c}Phi (n,P(ntau ^+)), end{aligned}$$
(11)

with (Phi (n,P(ntau ^+))= Phi ((n+1)tau ,R((n+1)tau ),P(ntau ^+))) and (P((n+1)tau )=P(ntau ^+)e^{-lambda tau }) for any (nge 0). In addition, evaluating the discrete component of system (4) at (t=(n+1)tau ), we obtain

$$begin{aligned} E_{r}((n+1)tau ^+)&= {} (1-kappa gamma )E_{r}((n+1)tau ), end{aligned}$$
(12)

$$begin{aligned} E_m((n+1)tau ^+)&= {} frac{{mathscr {B}}_0E_m((n+1)tau )}{{mathscr {B}}_0+gamma E_r((n+1)tau )}, end{aligned}$$
(13)

$$begin{aligned} P((n+1)tau ^+)= & {} left[ 1-mu +gamma E_{r}((n+1)tau )frac{E_{m}((n+1)tau ^+)}{e_{1/2}+E_{m}((n+1)tau ^+)}right] P((n+1)tau ). end{aligned}$$
(14)

Finally, substituting the Eqs. (10)–(11) into the Esq. (12)–(14) we obtain the discretization given by Eq. (6).
In order to obtain the equilibrium points of system (6), we can solve the following equations:

$$begin{aligned} e_{r}&= {} (1-kappa gamma )[e_{r}+alpha Phi (rho )],\ e_{m}&= {} dfrac{{mathscr {B}}_{0}[e_{m}+(1-alpha )xi _{c}Phi (rho )]}{{mathscr {B}}_0+gamma [e_{r}+alpha Phi (rho )]},\ rho&= {} left{ 1-mu +gamma [e_{r}+alpha Phi (rho )]dfrac{e_{m}}{e_{1/2}+e_{m}}right} rho e^{-lambda tau }, end{aligned}$$

where (e_{r}:=lim _{nrightarrow +infty }{E_{r}(ntau ^+)}), (e_{m}:=lim _{nrightarrow +infty }{E_{m}(ntau ^+)}), (rho :=lim _{nrightarrow +infty }{P(ntau ^+)}) and then, (Phi (rho ):=lim _{nrightarrow +infty }{Phi (n,P(ntau ^+))}). From the first equation, we have (e_{r}=(1-kappa gamma )alpha Phi (P)/kappa gamma ) and then (e_{m}=kappa {mathscr {B}}_0 (1-alpha )xi _{c}/alpha ) for (Pin {0,rho }) such that (Phi (0)=R_{max}(e^{lambda tau }-1)/lambda ) and

$$begin{aligned} Phi (rho )={mathscr {A}},quad text {where}quad {mathscr {A}}=dfrac{e^{lambda tau }-1+mu }{{mathscr {B}}_0}cdot left{ dfrac{e_{1/2}xi _{c}^{-1}}{1-alpha } +dfrac{kappa {mathscr {B}}_0}{alpha }right} . end{aligned}$$
(15)

Therefore, assuming (lambda =0), we have (Gamma (rho )=K-{mathscr {A}}rho ) (from equations (7) and (15)) where (Gamma :=lim _{nrightarrow +infty }{R((n+1)tau )}), (rho ) is the solution of

$$begin{aligned} r_0ln left( dfrac{K-{mathscr {A}}rho }{K}right) -{mathscr {A}}=-R_{max}tau , end{aligned}$$

if, and only if,

$$begin{aligned} rho =dfrac{K}{{mathscr {A}}}left[ 1-exp left( dfrac{{mathscr {A}}-R_{max}tau }{r_0}right) right] . end{aligned}$$
(16)

Long-term population dynamics
From the discretization (6), there are two dynamic behaviors for the long-term population abundance: extinction (see Fig. 2a) and persistence (see Fig. 2b).
Figure 2

Long-term behavior of solutions of the model (4). Peak values correspond to the solution of the discrete model (6) in its population component. (a) Extinction behavior, considering (alpha in {0.1,,0.8}) as energy allocation strategy toward reproductive and (b) persistence behavior considering (alpha in {0.35,, 0.55}). We consider the following parameter set (eta =(2,0.5,2,0.25,500,2,1,0.5,0.5,0.86,alpha ,0.1)) where the constant of fertility costs is described by (kappa =(1+gamma )^{-1}).

Full size image

The differentiation of these behaviors strongly depends on the individual consumption of resources defined throughout each non-reproductive season by

$$begin{aligned} C_{I}(t) = frac{[K-R(t)]e^{lambda (t-ntau )}}{P(ntau ^+)},, tin (ntau ,(n+1)tau ]. end{aligned}$$

At the end of each non-reproductive season, namely at the time ( t = (n + 1) tau ), the individual consumption is given by

$$begin{aligned} C_I((n+1)tau ) = frac{[K-R((n+1)tau )]e^{lambda tau }}{P(ntau ^+)}, end{aligned}$$
(17)

where (R((n+1)tau )) is the non-consumed resource density by the population, the amount that is obtained from the implicit solution of the resource density equation in the continuous component of the system (4). Thus, projecting the individual consumption of the resource into the long term, and taking (Gamma (rho )=(K-{mathscr {A}}rho +lambda {mathscr {I}})e^{-lambda tau }), the expression (17) assumes the form

$$begin{aligned} C_{I}^{infty }(rho )=left{ begin{array}{ll} {mathscr {A}}+dfrac{K(e^{lambda tau }-1)-lambda {mathscr {I}}}{rho },&{}quad text {if}, rho ne 0,\ &{}\ dfrac{R_{max}(e^{lambda tau }-1)}{lambda },&{}quad text {if}, rho =0, end{array}right. end{aligned}$$
(18)

where ({mathscr {I}}:=lim _{nrightarrow +infty }{int _{ntau }^{(n+1)tau }{R(s)e^{lambda (s-ntau )}ds}}). On the one hand, the individual consumption ( C_{I}^{infty } ) is composed of a basis amount corresponding to the term ({mathscr {A}} ) and an amount resulting from the equal division of a resource not consumed by individuals dying during the non-reproductive season, ( K(e^{lambda tau } -1) – lambda {mathscr {I}}). Furthermore, when the population experiences a reduced mortality during the non-reproductive season (i.e., ( lambda approx 0) ), the individual consumption is ( C_{I}^{infty } (rho ) approx {mathscr {A}} ). On the other hand, whether the population abundance is low, the per capita resource is high, implying an individual consumption close to (R_ {max} (e^{lambda tau } -1) / lambda ) (equivalent to taking the limit of ( C_{I}((n+1)tau )) as ( P(ntau ^+)rightarrow 0 )). Certainly, this quantity does not represent the effective individual consumption, but rather establishes an upper limit for this and therefore represents a value of non-persistence. Thus, behaviors related to the equilibrium solutions of the discrete system (6) can be differentiated by the threshold value

$$begin{aligned} {mathscr {U}}=dfrac{C_{I}^{infty }(0)}{C_{I}^{infty }(rho )}. end{aligned}$$

In particular, when the mortality of the population is low, the long-term abundance is described by Eq. (16) and the threshold value assumes the following form ({mathscr {U}}=R_{max}tau /{mathscr {A}}). Thus, we conclude that the persistence of the population is established when ({mathscr {U}} >1) and extinction when ({mathscr {U}}le 1).
Finally, we can see that the stabilization of population size in the long term is in response to a dense-dependent behaviour where the per capita growth rate in the long term is (r_{infty }:={alpha {mathscr {A}}/kappa }{mathscr {S}}(e_m)) equivalent to mortality fraction (mu ), where (e_{m}=kappa {mathscr {B}}_{0}(1-alpha )xi _{c}/alpha ) is the equilibrium value of energy maintenance after the reproductive season. In addition, the derivative of r with respect to P is

$$begin{aligned} dfrac{dr}{dP} = gamma dfrac{dPhi }{d P}left{ alpha {mathscr {S}}(E_{m})+(E_{r}+alpha Phi )dfrac{d{mathscr {S}}}{dE_{m}}cdot dfrac{dE_{m}}{dPhi }right} , end{aligned}$$

where (dPhi /d P1) then the long-term population behavior is persistence.

Proof
We divide the proof into two cases: (kappa gamma =1) and (0 More